DIFFERENTIAL EQUATIONS MTH401

Transcription

1 DIFFERENTIAL EQUATIONS MTH Virual Uivrsi of Pakisa Kowldg bod h boudaris

2 Tabl of Cos Iroduio... Fudamals.... Elms of h Thor.... Spifi Eampls of ODE s.... Th ordr of a quaio.... Ordiar Diffrial Equaio....5 Parial Diffrial Equaio....6 Rsuls from ODE daa....7 BVP Eampls Propris of ODE s Suprposiio Eplii Soluio Implii Soluio... 5 Sparabl Equaios Soluio sps of Sparabl Equaios Eris... Homogous Diffrial Equaios Mhod of Soluio Equaios rduibl o homogous form Cas Cas Eris... 5 Ea Diffrial Equaios Mhod of Soluio Eris Igraig Faor Thiqu Cas Cas Cas Cas... 9

3 6.5 Eris Firs Ordr Liar Equaios Mhod of soluio Eris... 8 Broulli Equaios Mhod of soluio Eris Subsiuios Eris Solvd Problms... 5 Appliaios of Firs Ordr Diffrial Equaios Orhogoal Trajoris Orhogoal urvs Orhogoal Trajoris (OT) Mhod of fidig Orhogoal Trajor Populaio Damis Radioaiv Da Nwo's Law of Coolig Carbo Daig... 8 Appliaios of No-liar Equaios Logisi quaio Soluio of h Logisi quaio Spial Cass of Logisi Equaio A Modifiaio of LE Chmial raios Misllaous Appliaios... 9 Highr Ordr Liar Diffrial Equaios Prlimiar hor Iiial -Valu Problm Soluio of IVP Thorm ( Eis ad Uiquss of Soluios)... 95

4 . Boudar-valu problm (BVP) Soluio of BVP Possibl Boudar Codiios Liar Dpd....6 Liar Idpd Cas of wo fuios....7 Wroskia....8 Thorm (Cririo for Liarl Idpd Fuios)....9 Eris... Soluios of Highr Ordr Liar Equaios Prlimiar Thor Suprposiio Priipl Liar Idpd of Soluios Fudamal S of Soluios Eis of a Fudamal S....5 Gral Soluio-Homogous Equaios....6 No-Homogous Equaios....7 Complmar Fuio....8 Gral Soluio of No-Homogous Equaios....9 Suprposiio Priipl for No-homogous Equaios.... Eris Cosruio of a Sod Soluio Gral Cas Ordr Rduio Eris... 6 Homogous Liar Equaios wih Cosa Coffiis Mhod of Soluio Cas (Disi Ral Roos) Cas (Rpad Roos) Cas (Compl Roos) Highr Ordr Equaios Cas (Ral disi roos)... 7

5 6.. Cas (Ral & rpad roos) Cas (Compl roos) Solvig h Auiliar Equaio Mhod of Udrmid Coffiis(Suprposiio Approah) Th form of Ipu fuio g () Soluio Sps Rsriio o Ipu fuio g Trial pariular soluios Ipu fuio g( ) as a sum Dupliaio bw p ad Eris... 8 Udrmid Coffii (Aihilaor Opraor Approah) Diffrial Opraors Diffrial Equaio i Trms of D Aihilaor Opraor Eris Udrmid Coffiis(Aihilaor Opraor Approah) Soluio Mhod Eris Variaio of Paramrs Firs ordr quaio Sod Ordr Equaio Summar of h Mhod Cosas of Igraio... 7 Variaio of Paramrs Mhod for Highr-Ordr Equaios Eris Appliaios of Sod Ordr Diffrial Equaio Simpl Harmoi Moio Hook s Law Nwo s Sod Law Wigh... 87

6 .. Diffrial Equaio Iiial Codiios Soluio ad Equaio of Moio Alraiv form of Soluio Ampliud A Vibraio or a Cl Priod of Vibraio Frqu Eris Dampd Moio Dampig For Th Diffrial Equaio Soluio of h Diffrial Equaio Alraiv form of h Soluio..... Quasi Priod Eris... 9 Ford Moio.... Ford moio wih dampig.... Trasi ad Sad-Sa Trms.... Moio wihou Dampig Elri Ciruis Th LRC Sris Ciruis Rsisor Iduor Capaior....6 Kirhhoff s Volag Law Th Diffrial Equaio Soluio of h diffrial quaio... Cas Ral ad disi roos... Cas Ral ad qual... Cas Compl roos... 5 Ford Moio (Eampls)... 6 Diffrial Equaios wih Variabl Coffiis...

7 6. Cauh- Eulr Equaio Mhod of Soluio Cas-I (Disi Ral Roos) Cas II (Rpad Ral Roos) Cas III (Cojuga Compl Roos) Eriss Cauh-Eulr Equaio (Alraiv Mhod of Soluio) Eriss... 8 Powr Sris (A Iroduio) Powr Sris Covrg ad Divrg Th Raio Ts Irval of Covrg Radius of Covrg Covrg a a Edpoi Absolu Covrg Powr Sris Rprsaio of Fuios Thorm Sris ha ar Idiall Zro Aali a a Poi Arihmi of Powr Sris Powr Sris Soluio of a Diffrial Equaio Eris Soluio abou Ordiar Pois Aali Fuio Ordiar ad sigular pois Polomial Coffiis Thorm (Eis of Powr Sris Soluio) No-polomial Coffiis Eris Soluios abou Sigular Pois Rgular ad Irrgular Sigular Pois Polomial Coffiis... 66

8 . Mhod of Frobius Frobius Thorm Cass of Idiial Roos Cas I (Roos o Diffrig b a Igr)... 7 Soluios abou Sigular Pois Mhod of Frobius (Cass II ad III) Cas II (Roos Diffrig b a Posiiv Igr) Bssl s Diffrial Equaio Sris Soluio of Bssl s Diffrial Equaio Bssl s Fuio of h Firs Kid Lgdr s Diffrial Equaio Lgdr s Polomials Rodrigus Formula for Lgdr s Polomials Graig Fuio For Lgdr s Polomials Rurr Rlaio Orhogoall of Lgdr s Polomials Normali odiio for Lgdr Polomials....7 Eris... 5 Ssms of Liar Diffrial Equaios Simulaous Diffrial Equaios Soluio of a Ssm Ssmai Elimiaio (Opraor Mhod) Ssms of Liar Diffrial Equaios Soluio of Usig Drmias Soluio Mhod Eris... 7 Ssms of Liar Firs-Ordr Equaio Th h Ordr Ssm Liar Normal Form Rduio of a Liar Diffrial Equaio o a Ssm Ssms Rdud o Normal Form Dgra Ssms Appliaios of Liar Normal Forms... 9

9 8 Iroduio o Maris Mari Rows ad Colums Ordr of a Mari Squar Mari Equali of mari Colum Mari Mulipl of maris Addiio of Maris Diffr of Maris Mulipliaio of Maris Mulipliaiv Idi Zro Mari Assoiaiv Law Disribuiv Law Drmia of a Mari Traspos of a Mari Mulipliaiv Ivrs of a Mari No-Sigular Maris Drivaiv of a Mari of fuios Igral of a Mari of Fuios Augmd Mari Elmar Row Opraios Th Gaussia ad Gauss-Jordo Mhods Eris Th Eigvalu problm Eigvalus ad Eigvors Th No-rivial soluio Eris Maris ad Ssms of Liar Firs-Ordr Equaios Mari form of a ssm Iiial Valu Problm Thorm: Eis of a uiqu Soluio... 6

10 . Suprposiio Priipl Liar Dpd of Soluio Vors Liar Idpd of Soluio Vors Eris Maris ad Ssms of Liar s -Ordr Equaios (Coiud) Thorm Fudamal s of soluio Thorm (Eis of a Fudamal S) Gral soluio No-homogous Ssms Pariular Igral Thorm Complmar fuio Gral soluio of a No homogous ssms Fudamal Mari Eris Homogous Liar Ssms Eigvalus ad Eigvors Cas (Disi ral igvalus) Cas (Compl igvalus) Thorm (Soluios orrspodig o ompl igvalus ) Thorm(Ral soluios orrspodig o a ompl igvalu) Eris... 9 Ral ad Rpad Eigvalus Eigvalu of muliplii m Mhod of soluio Eigvalu of Muliplii Two Eigvalus of Muliplii Thr No-Homogous Ssm.... Dfiiio.... Mari Noaio.... Mhod of Soluio.... Mhod of Udrmid Coffiis...

11 .. Th form of F ()..... Dupliaio of Trms....5 Variaio of Paramrs....6 Eris... 6

12 Diffrial Equaios (MTH) Iroduio Bakgroud Liar Quadrai Cubi m ab abd Ssms of Liar quaios ab lm Soluio? Equaio Diffrial Opraor d d Takig ai drivaiv o boh sids l From h pas Algbra Trigoomr Calulus Diffriaio Igraio Diffriaio Algbrai Fuios Trigoomri Fuios Logarihmi Fuios Epoial Fuios Ivrs Trigoomri Fuios Mor Diffriaio Sussiv Diffriaio Highr Ordr Libiz Thorm Appliaios Maima ad Miima Tag ad Normal Parial Drivaivs f() f(,) zf(,) Coprigh Virual Uivrsi of Pakisa

13 Diffrial Equaios (MTH) Igraio Rvrs of Diffriaio B pars B subsiuio B Parial Fraios Rduio Formula Frqul rquird Sadard Diffriaio formula Sadard Igraio Formula Diffrial Equaios Somhig Nw Mosl old suff Prsd diffrl Aalzd diffrl Applid Diffrl Coprigh Virual Uivrsi of Pakisa

14 Diffrial Equaios (MTH) Fudamals Dfiiio of a diffrial quaio. Classifiaio of diffrial quaios. Soluio of a diffrial quaio. Iiial valu problms assoiad o DE. Eis ad uiquss of soluios. Elms of h Thor Appliabl o: Chmisr Phsis Egirig Mdii Biolog Ahropolog Diffrial Equaio ivolvs a ukow fuio wih o or mor of is drivaivs Ordiar D.E. a fuio whr h ukow is dpd upo ol o idpd variabl Eampls of D.Eqs d 5 d d d. Spifi Eampls of ODE s du F ( ). Gu ( ), h growh quaio d d θ g si θ F ( ), h pdulum quaio d l d d ε ( ) d ( ) d d 5 d d u v u v u u u u, h va dr Pol quaio, d Coprigh Virual Uivrsi of Pakisa

15 Diffrial Equaios (MTH) d Q dq Q L R E (), h LCR osillaor quaio d d C () a p p v a Riai quaio d u() dp b () (),. Th ordr of a quaio Th ordr of h highs drivaiv apparig i h quaio. Ordiar Diffrial Equaio d d 5 d d u a If a quaio oais ol ordiar drivaivs of o or mor dpd variabls, w.r. a sigl variabl, h i is said o b a Ordiar Diffrial Equaio (ODE). For ampl h diffrial quaio d d 5 d d is a ordiar diffrial quaio..5 Parial Diffrial Equaio Similarl a quaio ha ivolvs parial drivaivs of o or mor dpd variabls w.r. wo or mor idpd variabls is alld a Parial Diffrial Equaio (PDE). For ampl h quaio u u a is a parial diffrial quaio..6 Rsuls from ODE daa Th soluio of a gral diffrial quaio:f(,,,..., ()) is dfid ovr som irval I havig h followig propris: () ad is firs drivaivs is for all i I so ha () ad is firs - drivas mus b oiuous i I () saisfis h diffrial quaio for all i I Gral Soluio all soluios o h diffrial quaio a b rprsd i his form for all osas Pariular Soluio oais o arbirar osas Iiial Codiio Boudar Codiio Iiial Valu Problm (IVP) Coprigh Virual Uivrsi of Pakisa

16 Diffrial Equaios (MTH) Boudar Valu Problm(BVP) IVP Eampls Th Logisi Equaio p ap bp wih iiial odiio p() p; for p h soluio is: p() a / (b (a b)-a(-)) Th mass-sprig ssm quaio (a / m) (k / m) g (F() / m).7 BVP Eampls Diffrial quaios 9 si() wih iiial odiios (), (p) - () (/8) si() os() si () p wih iiial odiios (), () - () os(p) ()si(p).8 Propris of ODE s Liar if h h-ordr diffrial quaio a b wri: a()() a-()(-)... a a() h() Noliar o liar ( )-( ) 5.9 Suprposiio Suprposiio allows us o dompos a problm io smallr, simplr pars ad h ombi hm o fid a soluio o h origial problm.. Eplii Soluio A soluio of a diffrial quaio d d d F,,,,, d d d ha a b wri as f() is kow as a plii soluio. Eampl: Th soluio is a plii soluio of h diffrial quaio. Implii Soluio d d d d A rlaio G(,) is kow as a implii soluio of a diffrial quaio, if i dfis o or mor plii soluio o I. Eampl: Th soluio - is a implii soluio of h quaio - / as i dfis wo plii soluios (-)/ Coprigh Virual Uivrsi of Pakisa 5

17 Diffrial Equaios (MTH) Sparabl Equaios Th diffrial quaio of h form d f (, ) d is alld sparabl if i a b wri i h form d h( ) g( ) d. Soluio sps of Sparabl Equaios To solv a sparabl quaio, w prform h followig sps:. W solv h quaio g ( ) o fid h osa soluios of h quaio.. For o-osa soluios w wri h quaio i h form. d h( ) d g( ) Th igra d h( ) d g( ) o obai a soluio of h form G ( ) H ( ) C. W lis h ir osa ad h o-osa soluios o avoid rpiio... If ou ar giv a IVP, us h iiial odiio o fid h pariular soluio. No ha: (a) No d o us wo osas of igraio baus C C C. (b) Th osas of igraio ma b rlabld i a ovi wa. () Si a pariular soluio ma oiid wih a osa soluio, sp is impora. Eampl : d Fid h pariular soluio of, () d Soluio:. B solvig h quaio:,w obai h osa soluios: ±. Rwri h quaio as d d Rsolvig io parial fraios ad igraig, w obai d d Coprigh Virual Uivrsi of Pakisa 6

18 Diffrial Equaios (MTH) Igraio of raioal fuios, w g l l C. Th soluios o h giv diffrial quaio ar l l C ±. Si h osa soluios do o saisf h iiial odiio, w plug i h odiio Wh i h soluio foud i sp o fid h valu of C. l C Th abov implii soluio a b rwri i a plii form as: Eampl : d d Solv h diffrial quaio Soluio:. W fid roos of h quaio o fid osa soluios; No osa soluios is baus h quaio has o ral roos.. For o-osa soluios, w spara h variabls ad igra d d / Si Thus / d / a ( a ( ) C So ha I is o as o fid h soluio i a plii form i.. as a fuio of.. Si o osa soluios, all soluios ar giv b h implii quaio foud i sp. Eampl : d Solv h iiial valu problm, ( ) d ) Coprigh Virual Uivrsi of Pakisa 7

19 Diffrial Equaios (MTH) Soluio:. Si ( )( ) Th quaio is sparabl & has o osa soluios baus o ral roos of.. For o-osa soluios w spara h variabls ad igra. d ( ) d d ( ) d a ( ) C Whih a b wri as a C. Si o osa soluios, all soluios ar giv b h implii or plii quaio.. Th iiial odiio ( ) givs π C a () Th pariular soluio o h iiial valu problm is a ( ) π π or i h plii form a Eampl : Solv ( ) d d Soluio: Dividig wih ( ), w a wri h giv quaio as d d ( ). Th ol osa soluio is. For o-osa soluio w spara h variabls Coprigh Virual Uivrsi of Pakisa 8

20 Diffrial Equaios (MTH) d Igraig boh sids, w hav d d d l l l l. ± or ( ) C C ± ( ), If w us l isad of h h soluio a b wri as l l l or l l ( ) So ha ( ).. Th soluios o h giv quaio ar Eampl 5 Solv d ( ) d. ( ) d Soluio: Th diffrial quaio a b wri as d. Si. Thrfor, h ol osa soluio is.. W spara h variabls d d or d ( ) d Igraig, wih us igraio b pars b pars o h firs rm, ilds 9 Coprigh Virual Uivrsi of Pakisa 9

21 Diffrial Equaios (MTH) ( ). All h soluios ar: ( ) whr 9 Eampl 6: Solv h iiial valu problms d d (a) ( ), () d, (). d (b) ( ) ad ompar h soluios. Soluios:. Si ( ). Thrfor, h ol osa soluio is.. W spara h variabls d ( ) d or Igraig boh sids w hav or ( ) ( ) ( - ) d d d d. All h soluios of h quaio ar. W plug i h odiios o fid pariular soluios of boh h problms (a) ( ) wh. So w hav Th pariular soluio is Coprigh Virual Uivrsi of Pakisa

22 Diffrial Equaios (MTH) So ha h soluio is, whih is sam as osa soluio. (b) ( ).. wh. So w hav. So ha soluio of h problm is 5. Compariso: A radial hag i h soluios of h diffrial quaio has Ourrd orrspodig o a vr small hag i h odiio!! Eampl 7: Solv h iiial valu problms d d (a) ( )., () d d (b) ( )., (). Soluio: (a) Firs osidr h problm d d ( )., () W spara h variabls o fid h o-osa soluios d (.) ( ) d Igra boh sids d ( ) (.) ( ) d So ha a.. Coprigh Virual Uivrsi of Pakisa

23 Diffrial Equaios (MTH) a. a.. ( ) [.( ) ] [ ] or. a. ( ) Applig ( ) wh a Thus h soluio of h problm is (b) Now osidr h problm ( ).( ) (. ). a d d, w hav ( )., (). W spara h variabls o fid h o-osa soluios d ( ) (.) d ( ) d ( ) (.) d l... Applig h odiio ( ) wh l.... l..... Coprigh Virual Uivrsi of Pakisa

24 Diffrial Equaios (MTH) Simplifiaio: B usig h propr Compariso: a b d. a b a b. d d Th soluios of boh h problms ar (a). a(. ) (b)... Agai a radial hag has ourrd orrspodig o a vr small i h diffrial quaio!. Eris Solv h giv diffrial quaio b sparaio of variabls.. d d 5 Coprigh Virual Uivrsi of Pakisa

25 Diffrial Equaios (MTH). s d s d. si d os ( ) d. d d 8 5. d d 6. ( ) d ( ) d d d 7. ( ) Solv h giv diffrial quaio subj o h idiad iiial odiio. 8. ( ) si d ( os )d, ( ), ( ) 9. ( ) d ( ) d. ( ) d d, ( ) Coprigh Virual Uivrsi of Pakisa

26 Diffrial Equaios (MTH) Homogous Diffrial Equaios A diffrial quaio of h form d d f (, ) Is said o b homogous if h fuio f (, ) is homogous, whih mas For som ral umbr, for a umbr. f (, ) f (, ) Eampl Drmi whhr h followig fuios ar homogous f (, ) g(, ) l Soluio: Th fuios f (, ) is homogous baus ( /( )) f (, ) f (, ) ( ) g(, w s ha g (, ) l l g(, ) ( ) Similarl, for h fuio ) Thrfor, h sod fuio is also homogous. H h diffrial quaios d d d d f (, ) g(, ) Ar homogous diffrial quaios Coprigh Virual Uivrsi of Pakisa 5

27 Diffrial Equaios (MTH). Mhod of Soluio d To solv h homogous diffrial quaio f (, ).W us h subsiuio d v.if f (, ) is homogous of dgr zro, h w hav f (, ) f (, v) F( v) dv Si v v, h diffrial quaio boms v f (, v) d This is a sparabl quaio. W solv ad go bak o old variabl hrough Summar:. Idif h quaio as homogous b hkig f (, ) f (, ) ;. Wri ou h subsiuio v ; v.. Through as diffriaio, fid h w quaio saisfid b h w fuio v ; dv v d f (, v). Solv h w quaio (whih is alwas sparabl) o fid v ; 5. Go bak o h old fuio hrough h subsiuio v ; 6. If w hav a IVP, w d o us h iiial odiio o fid h osa of igraio. Cauio: Si w hav o solv a sparabl quaio, w mus b arful abou h osa soluios. If h subsiuio v dos o rdu h quaio o sparabl form h h quaio is o homogous or somhig is wrog alog h wa. d 5 Eampl Solv h diffrial quaio d Soluio: Sp. I is as o hk ha h fuio fuio. Sp. To solv h diffrial quaio w subsiu f 5, ) ( is a homogous v Coprigh Virual Uivrsi of Pakisa 6

28 Diffrial Equaios (MTH) Sp. Diffriaig w.r., w obai whih givs dv d 5v v v v v 5v v This is a sparabl. A his sag plas rfr o h Cauio! 5v v Sp. Solvig b sparaio of variabls all soluios ar impliil giv b l( v ) l v l( ) C Sp 5. Goig bak o h fuio hrough h subsiuio v, w g ( ) ( ) l l l l l l l l l, l ( ) ( ) l l l ( ) ( ) l. l ( ) ( ). ( ) ( ) C No ha h implii quaio a b rwri as ( ) C. Equaios rduibl o homogous form Th diffrial quaio d d a ( b is o homogous. Howvr, i a b rdud o a homogous form as daild blow ) a b Coprigh Virual Uivrsi of Pakisa 7

29 Diffrial Equaios (MTH).. Cas a a b b W us h subsiuio a b whih rdus h quaio o a sparabl z quaio i h variabls ad z. Solvig h rsulig sparabl quaio ad rplaig z wih a b, w obai h soluio of h giv diffrial quaio... Cas a a b b I his as w subsiu X h, Y k Whr h ad k ar osas o b drmid. Th h quaio boms dy dx a X b Y a h b k a X b Y a h b k W hoos h ad k suh ha This rdus h quaio o a h b k a h b k dy dx a X by a X b Y Whih is homogous diffrial quaio i X ady, ad a b solvd aordigl. Afr havig solvd h las quaio w om bak o h old variabls ad. Eampl Solv h diffrial quaio d d Soluio: a b d dz Si, w subsiu z, so ha a b d d Coprigh Virual Uivrsi of Pakisa 8

30 Diffrial Equaios (MTH) Thus h quaio boms dz d z z i.. dz d z 7 z z This is a variabl sparabl form, ad a b wri as dz d z 7 Igraig boh sids w g z 9 l( z 7) A Simplifig ad rplaig z wih 9, w obai l( 7) A 9 ( ) A or ( ) 7, Eampl Solv h diffrial quaio d d ( ) 5 Soluio: B subsiuio X h, Y k, h giv diffrial quaio dy ( X Y ) ( h k ) rdus o dx ( X Y ) ( h k 5) W hoos h ad k suh ha h k, h k 5 Solvig hs quaios w hav h, k dy dx X Y X Y This is a homogous quaio. W subsiu X dv dx V V or V dv V dx X. Thrfor, w hav Y VX o obai Rsolvig io parial fraios ad igraig boh sids w obai ( V ) ( V ) dv dx X l or ( V ) l( V ) l X l A Simplifig ad rmovig ( l ) from boh sids, w g ( ) /( V ) CX C A V, Coprigh Virual Uivrsi of Pakisa 9

31 Diffrial Equaios (MTH) l ( V) l ( V) l X l A ( ) ( ) ( ) l( V ) l V l XA l( V ) V l XA ( V ) V XA akig powr " " o boh sids ( V) ( V) X A Y pu V X Y Y ( ) X A X X X Y X Y X A X X ( X Y) X X A X Y sa, A ( X Y) X Y pu X, Y ( ) / Now subsiuig ( ) /( ) C Y V, X, Y ad simplifig, w obai X.This is soluio of h giv diffrial quaio, a implii o.. Eris Solv h followig Diffrial Equaios. ( ) d d d. d. d d Coprigh Virual Uivrsi of Pakisa

32 Diffrial Equaios (MTH). d os d 5. ( ) d d Solv h iiial valu problms 6. ( 9 5 ) d ( 6 ) d, () 6 d, d 7. ( ) / / 8. ( ) d d, () d 9. osh, () d Coprigh Virual Uivrsi of Pakisa

33 Diffrial Equaios (MTH) 5 Ea Diffrial Equaios d f (, d L us firs rwri h giv diffrial quaio ) io h alraiv form M (, ) d N(, ) d whr M (, ) f (, ) N(, ) This quaio is a a diffrial quaio if h followig odiio is saisfid M N This odiio of ass isurs h is of a fuio F (, ) suh ha F F M (, ), N(, ) 5. Mhod of Soluio If h giv quaio is a h h soluio produr osiss of h followig sps: M N Sp. Chk ha h quaio is a b vrifig h odiio F Sp. Wri dow h ssm F M (, ), N(, ) Sp. Igra ihr h s quaio w. r. o or d w. r. o. If w hoos h s quaio h F(, ) M (, ) d θ ( ).Th fuio θ () fuio of, igraio w.r.o ; big osa. is a arbirar Sp. Us sod quaio i sp ad h quaio i sp o fid θ (). ( M (, ) d) ( ) N(, ) F θ θ ( ) N(, ) M (, ) d Sp 5. Igra o fid θ () ad wri dow h fuio F (, ); Sp 6. All h soluios ar giv b h implii quaio F (, ) C Sp 7. If ou ar giv a IVP, plug i h iiial odiio o fid h osa C. Coprigh Virual Uivrsi of Pakisa

34 Diffrial Equaios (MTH) Cauio: should disappar from θ () Eampl Solv ( ) d ( ) d. Ohrwis somhig is wrog! Soluio: Hr M ad N M N, M N. H h quaio is a. Th LHS of h quaio mus b a a f f diffrial i.. a fuio f (, ) suh ha M ad N Igraig s of hs quaios w. r.., hav f (, ) h( ), whr h() is h osa of igraio. Diffriaig h abov quaio w. r.. ad f usig d, w obai h ( ) N Comparig h ( ) is idpd of or igraig, w hav h ( ) Thus f (, ).H h gral soluio of h giv quaio is giv b f (, ) i...no ha w ould sar wih h d quaio f N o rah o h abov soluio of h giv quaio! Eampl Solv h iiial valu problm ( si os si ) d ( si os ) d., ( ). Soluio: Hr M si os si ad N si os M si os si, N si os si, Coprigh Virual Uivrsi of Pakisa

35 Diffrial Equaios (MTH) M This implis N Thus giv quaio is a.h hr iss a fuio f f (, ) suh ha si os si M ad f si os N Igraig s of hs w. r.., w hav f (, ) si os h( ), Diffriaig his quaio w. r.. subsiuig i f N si os h ( ) si os Ad h ( ) or h( ) H h gral soluio of h giv quaio is f (, ) i.. si os C, whr C. Now applig h iiial odiio ha wh,, w hav os si 9 is h rquird soluio. Eampl : Solv h DE ( ) ( ) os d os d Soluio:Th quaio is ihr sparabl or homogous. As M N (, ) os (, ) os M ad si os N H h giv quaio is a ad a fuio f (, ) is for whih M f (, ) ad N( ) f f f f, whih mas ha os ad os.l us sar wih h sod quaio i.. os.igraig boh sids w.r.o, w obai. No ha whil igraig w.r.o, f (, ) d os d d is rad as osa. Thrfor f (, ) si h( ) f h is a arbirar fuio of. From his quaio w obai ad qua i o M, Coprigh Virual Uivrsi of Pakisa

36 Diffrial Equaios (MTH) f os h ( ) os.so ha h ( ) h( ) C H o-paramr famil of soluio is giv b si d d Eampl Solv ( ) Soluio: Clarl M (, ) ad ( ) N, M N Th quaio is a ad a fuio ( ) f f, suh ha W igra firs of hs quaios o obai. f (, ) g( ) Hr g ( ) is a arbirar fuio. W fid f g f ad qua i o N (, ) ( ) g ( ) g( ) f ad Cosa of igraio d o o b iludd as h soluio is giv b f (, ) H a o-paramr famil of soluios is giv b Eampl 5 Solv h iiial valu problm ( os si ) d ( ) d, ( ) Soluio: As M (, ) os.si N(, ) ( ) M Thrfor h quaio is a ad a fuio ( ) f os.si N f, suh ha f ad ( ).Now igraig d of hs quaios w.r.. kpig osa, w obai f (, ) ( ) h( ) Diffria w.r.. ad qua h rsul o M (, ) Coprigh Virual Uivrsi of Pakisa 5

37 Diffrial Equaios (MTH) f h ( ) os si h ( ) os si Igraig w.r.o, w obai h( ) ( os )( si ) d os Thus a o paramr famil soluios of h giv diffrial quaio is ( ) os ( ) os rplad b C. Th iiial odiio wh ha,whr has b dmad, ha ( ) os ( ) os. Thus h soluio of h iiial valu problm is ( ) 5. Eris Drmi whhr h giv quaios is a. If so, plas solv.. ( si si ) d ( os os ) d. l d ( l )d. ( l ) d l d d. os si d 5. d d 6. Solv h giv diffrial quaios subj o idiad iiial odiios. 7. ( ) d ( ) d, () d 8., () 5 d d 9. os ( si ), () d so Coprigh Virual Uivrsi of Pakisa 6

38 Diffrial Equaios (MTH). Fid h valu of k, so ha h giv diffrial quaio is a. ( ) ( ) si k d si d. ( 6 os ) d ( k si ) d Coprigh Virual Uivrsi of Pakisa 7

39 Diffrial Equaios (MTH) 6 Igraig Faor Thiqu (, ) d N(, ) d If h quaio M is o a, h w mus hav M N.Thrfor, w look for a fuio u (, ) suh ha h quaio u (, ) M (, ) d u(, ) N(, ) d boms a. Th fuio u (, ) (if i iss) is alld h igraig faor (IF) ad i saisfis h quaio du o h odiio of ass. M u u M N u u N This is a parial diffrial quaio ad is vr diffiul o solv. Cosqul, h drmiaio of h igraig faor is rml diffiul p for som spial ass: Eampl Show ha /( ) is a igraig faor for h quaio ( ) d d, ad h solv h quaio. Soluio: Si M, N M N, M N ad h quaio is o a. Howvr, if h quaio is muliplid b /( ) h h quaio boms d d Now M ad N M N ( ) So ha his w quaio is a. Th quaio a b solvd. Howvr, i is simplr o obsrv ha h giv quaio a also wri d d l or d d[ l( )] ( ) d or d H, b igraio, w hav l k 6. Cas Wh a igraig faor u (), a fuio of ol. This happs if h prssio Coprigh Virual Uivrsi of Pakisa 8

40 Diffrial Equaios (MTH) M N N is a fuio of ol. Th h igraig faor u (, ) is giv b M N u p d N 6. Cas Wh a igraig faor u (), a fuio of ol. This happs if h prssio N M M N M u p d M 6. Cas is a fuio of ol. Th IF u (, ) is giv b M If h giv quaio is homogous ad Th 6. Cas N If h giv quaio is of h form f ( ) d g( ) d ad M N Th u M N u M N O h IF is foud, w mulipl h old quaio b u o g a w o, whih is a. Solv h a quaio ad wri h soluio. Advi: If possibl, w should hk whhr or o h w quaio is a? Summar: Sp. Wri h giv quaio i h form M (, ) d N(, ) d providd h quaio is o alrad i his form ad drmi M ad N. Sp. Chk for ass of h quaio b fidig whhr or o Coprigh Virual Uivrsi of Pakisa 9

41 Diffrial Equaios (MTH) M N Sp. (a) If h quaio is o a, h valua M N N If his prssio is a fuio of ol, h M N u( ) p d N Ohrwis, valua N M M If his prssio is a fuio of ol, h N M u( ) p d M I h abs of hs possibiliis, br us som ohr hiqu. Howvr, w ould also r ass ad i sp ad 5 Sp. Ts whhr h quaio is homogous ad If s h M u N M N Sp 5. Ts whhr h quaio is of h form f ( ) d g( ) d ad whhr M N If s h u M N Sp 6. Mulipl old quaio b u. if possibl, hk whhr or o h w quaio is a? Sp 7. Solv h w quaio usig sps dsribd i h prvious sio. Coprigh Virual Uivrsi of Pakisa

42 Diffrial Equaios (MTH) Eampl Solv h diffrial quaio Soluio: d d. Th giv diffrial quaio a b wri i form ( ) d ( ) d Thrfor M (, ) N (, ) M N. Now,. M N. To fid a IF w valua M N N whih is a fuio of ol..thrfor, a IF u () iss ad is giv b u( ) d l( ) 5. Muliplig h giv quaio wih h IF, w obai ( whih is a. (Plas hk!) ) d ( ) d 6. This sp osiss of solvig his las a diffrial quaio. Coprigh Virual Uivrsi of Pakisa

43 Diffrial Equaios (MTH) Soluio of w a quaio: M N. Si, h quaio is a.. W fid F (, ) b solvig h ssm F F. W igra h firs quaio o g. F(, ) θ ( ). W diffria F w. r.. ad us h sod quaio of h ssm i sp o obai 5. Igraig h las quaio o obai C F F (, ) W do' hav o kp h osa C, s sp. θ ( ) θ, No dpd o. θ. Thrfor, h fuio (, ) 6. All h soluios ar giv b h implii quaio F (, ) C i.. No ha i a b vrifid ha h fuio C u (, ) ( ) is aohr igraig faor for h sam quaio as h w quaio F is ( ) d ( ) d ( ) ( ) Coprigh Virual Uivrsi of Pakisa

44 Diffrial Equaios (MTH) is a. This mas ha w ma o hav uiquss of h igraig faor. Eampl. Solv ( ) d d Soluio: N M N M, N M Th quaio is o a.hr N N M Thrfor, I.F. is giv b d u p u Muliplig h quaio b I.F, w hav ( ) d d.this quaio is a. Th rquird Soluio is 8 Eampl Solv si d d Soluio: Hr N M N M N M, si, Th quaio is o a. Now M M N Coprigh Virual Uivrsi of Pakisa

45 Diffrial Equaios (MTH) Thrfor, h IF is d u ( ) p Muliplig h quaio b, w hav d ( si ) d or d d si d or d ( ) si d Igraig, w hav os si Whih is h rquird soluio Eampl Solv ( ) d ( ) d Soluio: Comparig wih Md Nd w s ha M ad N ( ) Si boh M ad N ar homogous. Thrfor, h giv quaio is homogous. Now M N H, h faor u is giv b u u M N Muliplig h giv quaio wih h igraig faor u, w obai. Coprigh Virual Uivrsi of Pakisa

46 Diffrial Equaios (MTH) d d Now M ad N ad hrfor M N Thrfor, h w quaio is a ad soluio of his w quaio is giv b l l C Eampl 5 Solv ( ) d ( ) d Soluio: Th giv quaio is of h form Now omparig wih f ( ) d g( ) d Md Nd W s ha M ( ) ad N ( ) Furhr M N Thrfor, h igraig faor u is Coprigh Virual Uivrsi of Pakisa 5

47 Diffrial Equaios (MTH) u, u M N Now muliplig h giv quaio b h igraig faor, w obai d d Thrfor, soluios of h giv diffrial quaio ar giv b l l C whr C C 6.5 Eris Solv b fidig a I.F. ( ) d d d si. d d. ( ) d ( ) d. ( ) d d 5. ( ) d d 6. ( ) d ( ) d d 7. d 8. ( ) d ( ) d 9. d ( ) d. ( ) si os d d Coprigh Virual Uivrsi of Pakisa 6

48 Diffrial Equaios (MTH) 7 Firs Ordr Liar Equaios Th diffrial quaio of h form: d a ( ) b( ) ( ) d is a liar diffrial quaio of firs ordr. Th quaio a b rwri i h followig famous form. whr p () ad q() d d ar oiuous fuios. p( ) q( ) 7. Mhod of soluio Th gral soluio of h firs ordr liar diffrial quaio is giv b u( ) q( ) d C u( ) Whr u ( ) p( p( ) d) u( Th fuio ) is alld h igraig faor. If i is a IVP h us i o fid h osa C. Summar:. Idif ha h quaio is s ordr liar quaio. Rwri i i h form d d if h quaio is o alrad i his form.. Fid h igraig faor. Wri dow h gral soluio p( ) q( ) u( ) p( ) d u( ) q( ) d C u( ). If ou ar giv a IVP, us h iiial odiio o fid h osa C. 5. Plug i h alulad valu o wri h pariular soluio of h problm. Coprigh Virual Uivrsi of Pakisa 7

49 Diffrial Equaios (MTH) Eampl : Solv h iiial valu problm Soluio:.Th quaio is alrad i h sadard form wih. Si a( ) os ( ), () d d p( ) q( ) p( ) a q() os a d l os l s Thrfor, h igraig faor is giv b u( ) a d s. Furhr, baus So ha h gral soluio is giv b s os d os d si C s si ( si C) os. W us h iiial odiio ( ) o fid h valu of h osa C ( ) C 5. Thrfor h soluio of h iiial valu problm is ( si ) os Coprigh Virual Uivrsi of Pakisa 8

50 Diffrial Equaios (MTH) Eampl : Solv h IVP d, d (). Soluio:.Th giv quaio is a s ordr liar ad is alrad i h rquisi form wih d d p( ) q( ) p( ) q( ). Si d l Thrfor, h igraig faor is giv b. H, h gral soluio is giv b Now d u( ) ( u( ) q( ) d C, u( ) u( ) q( ) d d ( ) ( ) d ( ) a d ) ( Th firs igral is larl. For h d w will us igraio b pars wih as firs fuio ad as ( ) d fuio. ( ( d d a ) d a ( ) a ( ) a ) - Th gral soluio is: ( ) a ( ) C. Th odiio ( ). givs C. 5. Thrfor, soluio o h iiial valu problm a b wri as: ) ( ) ( ) a ( ).( d ( ) ) Coprigh Virual Uivrsi of Pakisa 9

51 Diffrial Equaios (MTH) Eampl : Fid h soluio o h problm os si. π os., Soluio:. Th quaio is s ordr liar ad is o i h sadard form d p( ) q( ) d Thrfor w rwri h quaio as os si os si. H, h igraig faor is giv b os d l si u ( ) si si. Thrfor, h gral soluio is giv b si d C os si si Si si d d os si os a Thrfor a C si os C si s C s () Th iiial odiio ( π / ) implis C whih givs C. () Thrfor, h pariular soluio o h iiial valu problm is Eampl Solv ( ) s s d d d Soluio: W hav d Coprigh Virual Uivrsi of Pakisa

52 Diffrial Equaios (MTH) This quaio is o liar i. L us rgard as dpd variabl ad as idpd variabl. Th quaio ma b wri as d d Or d d, whih is liar i. IF p d p l d d Muliplig wih h IF, w g d d Igraig, w hav ( ) is h rquird soluio. d d Eampl 5 Solv ( ) ( ) Soluio: Th quaio a b rwri as Hr P( ). d d ( ) Thrfor, a igraig faor of h giv quaio is [ ( ) ] ( ) p d IF p l Muliplig h giv quaio b h IF,w g ( ) ( ) [ ] d ( ) d whih is h rquird soluio. 7. Eris Solv h followig diffrial quaios d d. Igraig boh sids, w obai ( ). d d Coprigh Virual Uivrsi of Pakisa

53 Diffrial Equaios (MTH). d d d d. ( o ) d d. ( ) ( ) 5. ( ) d d ( ) dr 6. r s θ osθ dθ d 7. d 8. d ( )d Solv h iiial valu problms d d 9. ( ), ( ) d d. ( ) ( ), ( ) Coprigh Virual Uivrsi of Pakisa

54 Diffrial Equaios (MTH) 8 Broulli Equaios A diffrial quaio ha a b wri i h form d p( ) q( ) d is alld Broulli quaio. 8. Mhod of soluio, For h quaio rdus o s ordr liar DE ad a b solvd aordigl. For, w divid h quaio wih ad h pu d d v p( ) o wri i i h form q( ) Diffriaig w.r.., w obai v ( ) Thrfor h quaio boms dv d ( ) p( ) v ( ) q( ) This is a liar quaio saisfid b v. O i is solvd, ou will obai h fuio v () If >, h w add h soluio o h soluios foud h abov hiqu. Summar.Idif h quaio as Broulli quaio. Fid. If, divid b d d p( ) q( ) ad subsiu; v Coprigh Virual Uivrsi of Pakisa

55 Diffrial Equaios (MTH). Through as diffriaio, fid h w quaio dv ( ) p( ) v ( ) q( ) d. This is a liar quaio. Solv h liar quaio o fid v.. Go bak o h old fuio hrough h subsiuio v ( ). 6. If >, h ilud o i h soluio. 7. If ou hav a IVP, us h iiial odiio o fid h pariular soluio. Eampl : Solv h quaio d d Soluio:. Th giv diffrial a b wri as d d whih is a Broulli quaio wih p ( ), q( ),. Dividig wih w g Thrfor w subsiu d d v. Diffriaig w.r.. w hav So ha h quaio rdus o d d dv d Coprigh Virual Uivrsi of Pakisa

56 Diffrial Equaios (MTH) dv v d. This is a liar quaio. To solv his w fid h igraig faor u () d u( ) Th soluio of h liar quaio is giv b Si ( ) u( ) q( ) d d u( ) v Thrfor, h soluio for v is giv b ( ) d v C C. To go bak o w subsiu. Thrfor h gral soluio of h giv DE is v ± ( C ) 5. Si >, w ilud h i h soluios. H, all soluios ar, ± ( C ) Eampl : Solv d d Soluio: I h giv quaio w idif ( ) ( ) Thus h subsiuio w givs dw w. d Th igraig faor for his liar quaio is P, q ad. Coprigh Virual Uivrsi of Pakisa 5

57 Diffrial Equaios (MTH) d d d l H [ w]. Igraig his lar form, w g Si w, w obai l w or w. or w For > h rivial soluio is a soluio of h giv quaio. I his ampl, is a sigular soluio of h giv quaio. Eampl Solv: d d Soluio: Dividig () b, h giv quaio boms () Pu Th () rdus o This is liar i v. d d dv d v or. d d v ( ) dv d () () ( ) ( ) I.F p p d l ( ) Muliplig () b ( ), w g Coprigh Virual Uivrsi of Pakisa 6

58 Diffrial Equaios (MTH) ( ) dv d ( ) 5 / ( ) / v d or ( ) v ( ) Igraig, w hav d or v ( ) ( ) ( ) v / or ( ) is h rquird soluio. 8. Eris Solv h followig diffrial quaios d d. l d d. d d. d. ( ) d d d 5. ( ) / / Coprigh Virual Uivrsi of Pakisa 7

59 Diffrial Equaios (MTH) d d 6. Solv h iiial-valu problms d d 7., ( ) / d d / 8., ( ) d, d 9. ( ) ( ) d, d. ( ) 8. Subsiuios Eampl Somims a diffrial quaio a b rasformd b mas of a subsiuio io a form ha ould h b solvd b o of h sadard mhods i.. Mhods usd o solv sparabl, homogous, a, liar, ad Broulli s diffrial quaio. A quaio ma look diffr from a of hos ha w hav sudid i h prvious lurs, bu hrough a ssibl hag of variabls prhaps a apparl diffiul problm ma b radil solvd. Alhough o firm ruls a b giv o h basis of whih hs subsiuio ould b sld, a workig aiom migh b: Tr somhig! I somims pas o b lvr. Th diffrial quaio ( ) d ( ) d is o sparabl, o homogous, o a, o liar, ad o Broulli. Howvr, if w sar a h quaio log ough, w migh b prompd o r h subsiuio u or d du ud Si u Coprigh Virual Uivrsi of Pakisa 8

60 Diffrial Equaios (MTH) Th quaio boms, afr w simplif u d ( u) du. w obai u l l u l / /, whr dsird, was rplad b. W a also rpla b if No: Th diffrial quaio i h ampl posssss h rivial soluio, bu h his fuio is o iludd i h o-paramr famil of soluio. Eampl Solv d d Soluio: 6. Th prs of h rm Si du d d d d d promps us o r u du Thrfor, h quaio boms: u 6 d du 6 or u d This quaio has h form of s ordr liar diffrial quaio d P( ) Q( ) d 6 wih P( ) ad Q( ) Thrfor, h igraig faor of h quaio is giv b Coprigh Virual Uivrsi of Pakisa 9

61 Diffrial Equaios (MTH) I.F d l d d Muliplig wih h IF givs [ ] u 6 Igraig boh sids, w obai Eampl Solv u or. Soluio: d d / If w l u Th h giv diffrial quaio a b simplifid o u u du d Igraig boh sids, w hav u u du d Usig h igraio b pars o LHS, w hav u u u or ( ) u u Whr - W h r-subsiu u Coprigh Virual Uivrsi of Pakisa 5

62 Diffrial Equaios (MTH) ad simplif o obai ( ) / Eampl Solv Soluio: If w l Th d d d d u du / d Th, h quaio rdus o du u d Whih is sparabl form. Sparaig h variabls, w obai du d u Igraig boh sids ilds u du d or Th osa is wri as u for ovi. Si u / Thrfor d d d d or Coprigh Virual Uivrsi of Pakisa 5

63 Diffrial Equaios (MTH) 8. Eris d d a Solv h diffrial quaios b usig a appropria subsiuio d ( ) d / ( ) d ( / ) d d s l (a ) d d ( si d d l d d d d d ) Coprigh Virual Uivrsi of Pakisa 5

64 Diffrial Equaios (MTH) Eampl : ' Soluio: 9 Solvd Problms d dw pu w h w d d dw w w w d w w dw w w d w d wdw Igraig w l l l l d d Coprigh Virual Uivrsi of Pakisa 5

65 Diffrial Equaios (MTH) d ( - ) Eampl : d d ( - ) Soluio: d pu w dw ( w - w) w d dw w w-w d dw w-w d dw d ( w- w) dw d ( w- w ) dw d w(- w ) pu w d W g d - -l - l l -l - l - (- ) - (- w) - (- / ) Coprigh Virual Uivrsi of Pakisa 5

66 Diffrial Equaios (MTH) Eampl : ( ) ( ) Soluio:( -) ( ) ( -) ( ) ( -) ( ) Igra w.r.. ' ' (, ) - ( ) Diffria w.r.. ' ' f d d d d Hr M ad N M N f f ad f h h'( ) Igra w.r.. '' h() -C Eampl : w w w w h'( ) N w ( / ) d d ( / ) ( / ) ( / ) ( / ) d Soluio: ( / ) d pu / w Afr subsiuio dw d w d w Igraig dw l l l w l l ( ) ( / ) ( ) Coprigh Virual Uivrsi of Pakisa 55

67 Diffrial Equaios (MTH) d Eampl 5: d l l d Soluio: d l l d d l l p ( ) ad q ( ) l l I. F p( d) l l Mulipl boh sid b l d l d d ( l ) d Igra l Coprigh Virual Uivrsi of Pakisa 56

68 Diffrial Equaios (MTH) Eampl 6: ( ) - Soluio:Hr M N - M, d d N - M N Clarl Th giv quaio is o a. divid h quaio b o mak i a d - d M N Now - Equaio is a f f - Igra w.r.. ' ' f(,) Coprigh Virual Uivrsi of Pakisa 57

69 Diffrial Equaios (MTH) Eampl 7: d os ( si os ) d d Soluio: os ( si os ) d d si os d os os d [ a / ] d os I. F p( (a / ) d) s d s s s [ a / ] d os d s [ s a s ] s d d [ s ] s d s a Coprigh Virual Uivrsi of Pakisa 58

70 Diffrial Equaios (MTH) d l Eampl 8: d d l Soluio: d pu u d du d d du l u d du l u d Hr p( ) / Ad Q( ) I. F p( d) du u l d l d ( u ) l d Igra u [l-] [ l - ] Coprigh Virual Uivrsi of Pakisa 59

71 Diffrial Equaios (MTH) d Eampl 9: l d d Soluio: l d d l d pu l u du u d d IF.. d ( u ) d Igra. u l Coprigh Virual Uivrsi of Pakisa 6

72 Diffrial Equaios (MTH) d Eampl : s - l a d d Soluio:s - l a d pu l a u d du si os d d si os du -u si os d du -u d du u d I. F p( / d) du u d d ( u ) d u u l a - - Coprigh Virual Uivrsi of Pakisa 6

73 Diffrial Equaios (MTH) d Eampl : ( ) d d Soluio: ( ) d Pu u du u u d du u u d du u d u pu/ u w (Brouli's) - dw w d dw - w - d I. F p( - d) dw -w - d d - ( w ) - d Igra u - - w - Coprigh Virual Uivrsi of Pakisa 6

74 Diffrial Equaios (MTH) d Eampl : ( ) d d Soluio: ( ) d pu u wg du - u d du u d du u Igra d - u a - u a u a( ) a( ) Coprigh Virual Uivrsi of Pakisa 6

75 Diffrial Equaios (MTH) d Eampl : ( ) a d d Soluio :( ) a d pu u du u ( -) a d du u -u a d u du d u a Igra u a - a du d u a a (- ) du d u a - u u- aa a ( ) - a a - a Coprigh Virual Uivrsi of Pakisa 6

76 Diffrial Equaios (MTH) Eampl : Soluio : pu d d d d u du - u d du u d I. F Ep( d) du u d d ( u ) d Igraig u - Coprigh Virual Uivrsi of Pakisa 65

77 Diffrial Equaios (MTH) -( ) Eampl 5 : ' si -( ) Soluio : ' si pu u du -u si d du si d -u u du si d Igra u -os u l -os l -os Coprigh Virual Uivrsi of Pakisa 66

78 Diffrial Equaios (MTH) Eampl 6 : ' - Soluio : ' - pu u d d d du - d d d du - d d du - d du 6-9/ d Igra - 9l d u - 9l du Coprigh Virual Uivrsi of Pakisa 67

79 Diffrial Equaios (MTH) Eampl 7:Solvos( ) d d Soluio:os( ) d d d dv pu v or, w g d d dv os v[ -] d os v d dv [- ] dv os v os v v d [- s ] dv Igra v v- a v- a Coprigh Virual Uivrsi of Pakisa 68

80 Diffrial Equaios (MTH) Appliaios of Firs Ordr Diffrial Equaios I ordr o rasla a phsial phomo i rms of mahmais, w sriv for a s of quaios ha dsrib h ssm adqual. This s of quaios is alld a Modl for h phomo. Th basi sps i buildig suh a modl osis of h followig sps: Sp : W larl sa h assumpios o whih h modl will b basd. Ths assumpios should dsrib h rlaioships amog h quaiis o b sudid. Sp : Compll dsrib h paramrs ad variabls o b usd i h modl. Sp : Us h assumpios (from Sp ) o driv mahmaial quaios rlaig h paramrs ad variabls (from Sp ). Th mahmaial modls for phsial phomo of lad o a diffrial quaio or a s of diffrial quaios. Th appliaios of h diffrial quaios w will disuss i wo lurs ilud: Orhogoal Trajoris. Populaio damis. Radioaiv da. Nwo s Law of oolig. Carbo daig. Chmial raios... Orhogoal Trajoris W kow ha ha h soluios of a s ordr diffrial quaio,.g. sparabl quaios, ma b giv b a implii quaio F (,, C) wih paramr C, whih rprss a famil of urvs. Mmbr urvs a b obaid b fiig h paramr C. Similarl a h ordr DE will ilds a -paramr famil of urvs/soluios. (, C, C,, ), C F Th qusio ariss ha whhr or o w a ur h problm aroud: Sarig wih a -paramr famil of urvs, a w fid a assoiad h ordr diffrial quaio fr of paramrs ad rprsig h famil. Th aswr i mos ass is s. L us r o s, wih rfr o a -paramr famil of urvs, how o prod if h aswr o h qusio is s. Coprigh Virual Uivrsi of Pakisa 69

81 Diffrial Equaios (MTH) d. Diffria wih rsp o, ad g a quaio-ivolvig,, ad C. d. Usig h origial quaio, w ma b abl o limia h paramr C from h w quaio.. Th sp is doig som algbra o rwri his quaio i a plii form d d f (, ) Eampl Fid h diffrial quaio saisfid b h famil C Soluio:. W diffria h quaio wih rsp o, o g d C d. Si w hav from h origial quaio ha C h w g d d. Th plii form of h abov diffrial quaio is d d This las quaio is h dsird DE fr of paramrs rprsig h giv famil. Eampl. L us osidr h ampl of h followig wo familis of urvs Coprigh Virual Uivrsi of Pakisa 7

82 Diffrial Equaios (MTH) m C Th firs famil dsribs all h sraigh lis passig hrough h origi whil h sod famil dsribs all h irls rd a h origi. If w draw h wo familis oghr o h sam graph w g Clarl whvr o li irss o irl, h ag li o h irl (a h poi of irsio) ad h li ar prpdiular i.. orhogoal o ah ohr. W sa ha h wo familis of urvs ar orhogoal a h poi of irsio.. Orhogoal urvs A wo urvs C ad C ar said o b orhogoal if hir ag lis T ad T a hir poi of irsio ar prpdiular. This mas ha slops ar gaiv riproals of ah ohr, p wh T ad T ar paralll o h oordia as.. Orhogoal Trajoris (OT) I : ) aohr famil I (,, ) Wh all urvs of a famil G(,, orhogoall irs all urvs of : H h ah urv of h familis is said o b orhogoal rajor of h ohr. Coprigh Virual Uivrsi of Pakisa 7

83 Diffrial Equaios (MTH) Eampl: As w a s from h prvious figur ha h famil of sraigh lis famil of irls C ar orhogoal rajoris. m ad h Orhogoal rajoris our aurall i ma aras of phsis, fluid damis, i h sud of lrii ad magism. For ampl h lis of for ar prpdiular o h quipoial urvs i.. urvs of osa poial... Mhod of fidig Orhogoal Trajor Cosidr a famil of urvs I. Assum ha a assoiad DE ma b foud, whih is giv b: d Si d d d f (, ) givs slop of h ag o a urv of h famil I hrough (, ). Thrfor, h slop of h li orhogoal o his ag is. So ha h f (, ) slop of h li ha is ag o h orhogoal urv hrough (, ) is giv b. I ohr words, h famil of orhogoal urvs ar soluios o h f (, ) diffrial quaio d d f (, ) Th sps a b summarizd as follows: Summar: I ordr o fid Orhogoal Trajoris of a famil of urvs I w prform h followig sps: Sp. Cosidr a famil of urvs I ad fid h assoiad diffrial quaio. Sp. Rwri his diffrial quaio i h plii form d d f (, ) Sp. Wri dow h diffrial quaio assoiad o h orhogoal famil d d f (, ) Sp. Solv h w quaio. Th soluios ar al h famil of orhogoal urvs. Coprigh Virual Uivrsi of Pakisa 7

84 Diffrial Equaios (MTH) Sp 5. A spifi urv from h orhogoal famil ma b rquird, somhig lik a IVP. Eampl Fid h orhogoal Trajor o h famil of irls C Soluio: Th giv quaio rprss a famil of ori irls rd a h origi. Sp. W diffria w.r.. o fid h DE saisfid b h irls. d d Sp. W rwri his quaio i h plii form d d Sp. N w wri dow h DE for h orhogoal famil d d ( / ) Sp.This is a liar as wll as a sparabl DE. Usig h hiqu of liar quaio, w fid h igraig faor u( ) whih givs h soluio or d. u( ) m m m u( ) Whih rprs a famil of sraigh lis hrough origi. H h famil of sraigh lis m ad h famil of irls C ar Orhogoal Trajoris. Sp 5. A gomrial viw of hs Orhogoal Trajoris is: Coprigh Virual Uivrsi of Pakisa 7

85 Diffrial Equaios (MTH) Eampl Fid h Orhogoal Trajor o h famil of irls C Soluio:. W diffria h giv quaio o fid h DE saisfid b h irls. d d C, C. Th plii diffrial quaio assoiad o h famil of irls is d d. H h diffrial quaio for h orhogoal famil is Coprigh Virual Uivrsi of Pakisa 7

86 Diffrial Equaios (MTH) d d. This DE is a homogous, o solv his quaio w subsiu v / or quivall d d v. Th w hav dv v v d ad v Thrfor h homogous diffrial quaio i sp boms dv v v d v Algbrai maipulaios rdu his quaio o h sparabl form: dv d Th osa soluios ar giv b v v v v v v ( v Th ol osa soluio is v. ) To fid h o-osa soluios w spara h variabls Igra v dv d v v v d v d v v Rsolvig io parial fraios h igrad o LHS, w obai v v v v v( v v d v v v ) v v v H w hav v v v H h soluio of h sparabl quaio boms whih is quival o l v l[ v d v l v l[ v ] l lc ] Coprigh Virual Uivrsi of Pakisa 75

87 Diffrial Equaios (MTH) v C v whr C. H all h soluios ar v v C v W go bak o o g ad C m whih is quival o 5. Whih is -ais ad a famil of irls rd o -ais. A gomrial viw of boh h familis is show i h slid. Coprigh Virual Uivrsi of Pakisa 76

88 Diffrial Equaios (MTH). Populaio Damis bsom aural qusios rlad o populaio problms ar h followig: Wha will h populaio of a rai our afr.g. ars? How ar w proig h rsours from iio? Th asis populaio damis modl is h poial modl. This modl is basd o h assumpio: Th ra of hag of h populaio is proporioal o h isig populaio. If () miod assumpio w a wri P masurs h populaio of a spis a a im h baus of h abov Coprigh Virual Uivrsi of Pakisa 77

89 Diffrial Equaios (MTH) dp d kp whr h ra k is osa of proporioali. Clarl h abov quaio is liar as wll as sparabl. To solv his quaio w mulipl h quaio wih h igraig faor k o obai d d P k Igraig boh sids w obai P k C or P k C If P is h iiial populaio h P ( ) P. So ha P ( ) P k Clarl, w mus hav k > for growh ad < C P ad obai k for h da. Eampl: Th populaio of a rai ommui is kow o iras a a ra proporioal o h umbr of popl prs a a im. Th populaio has doubld i 5 ars, how log would i ak o ripl?. If i is kow ha h populaio of h ommui is, afr ars. Wha was h iiial populaio? Wha will b h populaio i ars? Soluio: Suppos ha P is iiial populaio of h ommui ad P () h populaio a a im h h populaio growh is govrd b h diffrial quaio dp kp d As w kow soluio of h diffrial quaio is giv b P ( ) P k P ( 5) P. Thrfor, from h las quaio w hav Si Coprigh Virual Uivrsi of Pakisa 78

90 Diffrial Equaios (MTH) This mas ha 5k 5k P P k l.695 or k Thrfor, h soluio of h quaio boms P( ) P.86 If is h im ak for h populaio o ripl h P l ars.86 P Now usig h iformaio P ( ),, w obai from h soluio ha (.86 )(), P P Thrfor, h iiial populaio of h ommui was P 6598 H soluio of h modl is P( ) So ha h populaio i ars is giv b,.589 ()(.86 ) P ( ) or P ( ) ( 6598)( 6.) or P ( ) Coprigh Virual Uivrsi of Pakisa 79

91 Diffrial Equaios (MTH) Radioaiv Da I phsis a radioaiv subsa disigras or rasmus io h aoms of aohr lm. Ma radioaiv marials disigra a a ra proporioal o h amou prs. Thrfor, if A () is h amou of a radioaiv subsa prs a im, h h ra of hag of A() wih rsp o im is giv b da ka d whr k is a osa of proporioali. L h iiial amou of h marial b A h A ( ) A. As disussd i h populaio growh modl h soluio of h diffrial quaio is ( ) A k A Th osa k a b drmid usig half-lif of h radioaiv marial. Th half-lif of a radioaiv subsa is h im i aks for o-half of h aoms i a iiial amou A o disigra or rasmu io aoms of aohr lm. Th halflif masurs sabili of a radioaiv subsa. Th logr h half-lif of a subsa, h mor sabl i is. If T dos h half-lif h A ( T ) Thrfor, usig his odiio ad h soluio of h modl w obai A A k So ha kt l A Thrfor, if w kow T, w a g k ad vi-vrsa. Th half-lif of som impora radioaiv marials is giv i ma books of Phsis ad Chmisr. For ampl h half-lif of C is 5568 ± ars. Eampl : A radioaiv isoop has a half-lif of 6 das. W hav g a h d of das. How muh radioisoop was iiiall prs? Soluio: L A () b h amou prs a im ad A h iiial amou of h isoop. Th w hav o solv h iiial valu problm. da ka, A() d Coprigh Virual Uivrsi of Pakisa 8

92 Diffrial Equaios (MTH) W kow ha h soluio of h IVP is giv b ( ) A k A If T h half-lif h h osa is giv k b l kt l or k T A ( ) Now usig h odiio, w hav A k So ha h iiial amou is giv b A l k 6 l 6. g Eampl A brdr raor ovrs h rlaivl sabl uraium 8 io h isoop pluoium 9. Afr 5 ars i is drmid ha.% of h iiial amou A of h pluoium has disigrad. Fid h half-lif of his isoop if h ra of disigraio is proporioal o h amou rmaiig. Soluio: L A () dos h amou rmaiig a a im, h w d o fid soluio o h iiial valu problm whih w kow is giv b da d ka, A() A ( ) A k A If.% disigraio of h aoms of A mas ha % of h subsa rmais. Furhr % of A quals (.99957) A. So ha A ( 5) (.99957) A So ha A 5k (.99957) A 5 k l(.99957) l(.99957) Or k A ) A.867 H ( Coprigh Virual Uivrsi of Pakisa 8

93 Diffrial Equaios (MTH) If T dos h half-lif h A A ( T ). Thus A A T.867 or T l l l T,8 ars.867 T. Nwo's Law of Coolig From primal obsrvaios i is kow ha h mpraur T () of a obj hags a a ra proporioal o h diffr bw h mpraur i h bod ad h mpraur T m of h surroudig virom. This is wha is kow as Nwo's law of oolig. If iiial mpraur of h oolig bod is T h w obai h iiial valu problm dt d k ( T T ), T () T m whr k is osa of proporioali. Th diffrial quaio i h problm is liar as wll as sparabl. Sparaig h variabls ad igraig w obai dt k d T Tm This mas ha T ( ) l T Tm k C T T m k C k Now applig h iiial odiio soluio of h iiial valu problm is giv b whr Tm C C T ( ) Tm ( T T ( ) T, w s ha m T m ) H, If mpraurs a ims ad ar kow h w hav k k T ( T T T T ) m ( m ), ( ) m ( m ) T k T T C C T T. Thus h Coprigh Virual Uivrsi of Pakisa 8

94 Diffrial Equaios (MTH) So ha w a wri T ( T ( ) T ) T m m k( ) This quaio provids h valu of k if h irval of im is kow ad vivrsa. Eampl : Suppos ha a dad bod was disovrd a midigh i a room wh is mpraur was 8 F. Th mpraur of h room is kp osa a 6 F. Two hours lar h mpraur of h bod droppd o 75 F. Fid h im of dah. Soluio: Assum ha h dad prso was o sik, h o T ( ) 98.6 F T ad T 6 Thrfor, w hav o solv h iiial valu problm dt d ( T 6 ), () k T W kow ha h soluio of h iiial valu problm is k So ha T ( ) Tm ( T Tm ) T ( ( ) ) T k m T ( ) T m Th obsrvd mpraurs of h oolig obj, i.. h dad bod, ar Subsiuig hs valus w obai o T ( ) 8 F ad T ( ) 8 6 k as hours 75 6 So k l. 8 Now suppos ha ad do h ims of dah ad disovr of h dad bod h m o F 75 o o T ( ) T () 98.6 F ad T ( ) 8 F d ) T k( ) m kd ) T 8 6 For h im of dah, w d o drmi h irval T ( T ( m o F. Now Coprigh Virual Uivrsi of Pakisa 8

95 Diffrial Equaios (MTH) 8.6 or d l. 57 k H h im of dah is 7: PM.. Carbo Daig Th isoop C is produd i h amosphr b h aio of osmi radiaio o irog. Th raio of C- o ordiar arbo i h amosphr appars o b osa. Th proporioa amou of h isoop i all livig orgaisms is sam as ha i h amosphr. Wh a orgaism dis, h absorpio of C b brahig or aig ass. Thus ompariso of h proporioa amou of C prs, sa, i a fossil wih osa raio foud i h amosphr provids a rasoabl sima of is ag. Th mhod has b usd o da wood furiur i Egpia ombs. Si h mhod is basd o h kowldg of half-lif of h radio aiv C (56 ars approimal), h iiial valu problm disussd i h radioaivi modl govrs his aalsis. Eampl A fossilizd bo is foud o oai / of h origial amou of C. Drmi h ag of h fissil. Soluio: L A() b h amou prs a a im ad A h origial amou of C. Thrfor, h pross is govrd b h iiial valu problm. da ka, A() A d W kow ha h soluio of h problm is A( ) A k Si h half lif of h arbo isoop is 56 ars. Thrfor, So ha A A (56) A A 56k or 56k k.78 l Coprigh Virual Uivrsi of Pakisa 8

96 Diffrial Equaios (MTH) H A ) ( (.78) A If dos h im wh fossilizd bo was foud h A ( ) A (.78) A.78 l Thrfor l.78 55,8 ars A Coprigh Virual Uivrsi of Pakisa 85

97 Diffrial Equaios (MTH) Appliaios of No-liar Equaios As w kow ha h soluio of h poial modl for h populaio growh is P ) P k ( P big h iiial populaio. From his soluio w olud ha (a) If k > h populaio grows ad pad o ifii i.. lim P ( ) (b) If k < h populaio will shrik o approah, whih mas iio. No ha: () Th prdiio i h firs as ( k > ) diffrs subsaiall from wha is auall obsrvd, populaio growh is vuall limid b som faor! () Drimal ffs o h virom suh as polluio ad ssiv ad ompiiv dmads for food ad ful. a hav ihibiiv ffs o h populaio growh.. Logisi quaio Aohr modl was proposd o rmd his flaw i h poial modl. This is alld h logisi modl (also alld Vrhuls-Parl modl). Suppos ha a > is osa avrag ra of birh ad ha h dah ra is proporioal o h populaio P () a a im. Thus if pr idividual h dp a bp P d or dp d dp P d P( a bp ) is h ra of growh whr b is osa of proporioali. Th rm bp, b > a b irprd as ihibiio rm. Wh b, h quaio rdus o h o i poial modl. Soluio o h logisi quaio is also vr impora i ologial, soiologial ad v i maagrial sis... Soluio of h Logisi quaio Th logisi quaio dp P( a bp ) d a b asil idifid as a oliar quaio ha is sparabl. Th osa soluios of h quaio ar giv b P ( a bp ) Coprigh Virual Uivrsi of Pakisa 86

98 Diffrial Equaios (MTH) a P ad P b For o-osa soluios w spara h variabls dp d P( a bp) Rsolvig io parial fraios w hav / a b / a dp d P a bp Igraig l P l a bp C a a P l a ac a bp P a or C whr a bp Eas algbrai maipulaios giv ac a P( ) bc a bc C ac ac a Hr C is a arbirar osa. If w ar giv h iiial odiio P ( ) P, w obai obai Clarl No ha C P a P b. Subsiuig his valu i h las quaio ad simplifig, w a bp P( ) bp ap ( a bp ) a ap a lim P( ), limid growh bp b a P is a sigular soluio of h logisi quaio. b.. Spial Cass of Logisi Equaio.... Epidmi Sprad Suppos ha o prso ifd from a oagious disas is irodud i a fid populaio of popl. Coprigh Virual Uivrsi of Pakisa 87

99 Diffrial Equaios (MTH) d Th aural assumpio is ha h ra of sprad of disas is proporioal o h d umbr ) Th ( of h ifd popl ad umbr () d k d Si Thrfor, w hav h followig iiial valu problm d d k( ), () of popl o ifd popl. Th las quaio is a spial as of h logisi quaio ad has also b usd for h sprad of iformaio ad h impa of advrisig i rs of populaio... A Modifiaio of LE A modifiaio of h oliar logisi diffrial quaio is h followig dp d P( a b l P ) has b usd i h sudis of solid umors, i auarial prdiios, ad i h growh of rvu from h sal of a ommrial produ i addiio o growh or dli of populaio. Eampl Suppos a sud arrig a flu virus rurs o a isolad ollg ampus of suds. If i is assumd ha h ra a whih h virus sprads is proporioal o ol o h umbr of ifd suds bu also o h umbr of suds o ifd, drmi h umbr of ifd suds afr 6 das if i is furhr obsrvd ha afr das () 5. Soluio Assum ha o o lavs h ampus hroughou h duraio of h disas. W mus solv h iiial-valu problm d d k( ), (). Coprigh Virual Uivrsi of Pakisa 88

100 Diffrial Equaios (MTH) W idif a k ad b k Si h soluio of logisi quaio is Thrfor w hav ( ) P( ) bp k k 999k k Now, usig () 5, w drmi k ap ( a bp ) a 999 k k 9 W fid k l Thus Fiall. Chmial raios ( ) ( 6) 76 suds I a firs ordr hmial raio, h moluls of a subsa A dompos io smallr moluls. This domposiio aks pla a a ra proporioal o h amou of h firs subsa ha has o udrgo ovrsio. Th disigraio of a radioaiv subsa is a ampl of h firs ordr raio. If X is h rmaiig amou of h subsa A a a im h dx k d k < baus X is drasig. X Coprigh Virual Uivrsi of Pakisa 89

101 Diffrial Equaios (MTH) I a d ordr raio wo hmials A ad B ra o form aohr hmial C a a ra proporioal o h produ of h rmaiig oraios of h wo hmials. If X dos h amou of h hmial C ha has formd a im. Th h isaaous amous of h firs wo hmials A ad B o ovrd o h hmial C ar hmial C is giv b α X ad β X, rspivl. H h ra of formaio of dx d ( α - X)( X ) k β whr k is osa of proporioali. Eampl: A ompoud C is formd wh wo hmials A ad B ar ombid. Th rsulig raio bw h wo hmials is suh ha for ah gram of A, grams of B ar usd. I is obsrvd ha grams of h ompoud C ar formd i mius. Drmi h amou of C a a im if h ra of raio is proporioal o h amous of A ad B rmaiig ad if iiiall hr ar 5 grams of A ad grams of B. How muh of h ompoud C is prs a 5 mius? Irpr h soluio as Soluio: If X () do h umbr of grams of hmial C prs a a im. Th X ( ) ad X ( ) Suppos ha hr ar grams of h ompoud C ad w hav usd a grams of A ad b grams of B h Solvig h wo quaios w hav a b ad b a 8 a (/ 5) ad b ( / 5) 5 5 I gral, if hr wr for X grams of C h w mus hav a X 5 ad b 5 X Coprigh Virual Uivrsi of Pakisa 9

102 Diffrial Equaios (MTH) Thrfor h amous of A ad B rmaiig a a im ar h rspivl. X 5 ad 5 5 X Thrfor, h ra a whih hmial C is formd saisfis h diffrial quaio or dx d dx d W ow solv his diffrial quaio. X λ 5 X 5 5 k( 5 X )( X ), k λ / 5 B sparaio of variabls ad parial fraio, w a wri dx X ( 5 )( X ) kd / 5 X dx / X dx kd 5 X l X k Wh, w fid 5 X k Whr X, X, so i follows a his poi ha 5/. Usig 88 k l.58 5 Wih his iformaio w solv for X : X a Coprigh Virual Uivrsi of Pakisa 9

103 Diffrial Equaios (MTH).58 X ( ) 5.58 I is lar ha as.58 as. Thrfor X as. This fa a also b vrifid from h followig abl ha X as X This mas ha hr ar grams of ompoud C formd, lavig 5 () 5 ad () 5. Misllaous Appliaios grams of hmial A grams of hmial Th vloi v of a fallig mass m, subjd o air rsisa proporioal o isaaous vloi, is giv b h diffrial quaio dv m mg kv d Hr k > is osa of proporioali. Th ra a whih a drug dissmias io bloodsram is govrd b h diffrial quaio d A B d Hr A, B ar posiiv osas ad () dsribs h oraio of drug i h bloodsram a a im. Th ra of mmorizaio of a subj is giv b B Coprigh Virual Uivrsi of Pakisa 9

104 Diffrial Equaios (MTH) da d k ( M A k ) A Hr k >, k > ad A () is h amou of marial mmorizd i im, M is h oal amou o b mmorizd ad M A is h amou rmaiig o b mmorizd. Coprigh Virual Uivrsi of Pakisa 9

105 Diffrial Equaios (MTH) Highr Ordr Liar Diffrial Equaios. Prlimiar hor A diffrial quaio of h form a d d d ( ) a ( ) a( ) a ( ) g( ) d d d ( ) ( ) or a ( ) a( ) a( ) a ( ) g( ) whr a ( ), a( ),, a ( ), g( ) ar fuios of ad a ( ), is alld a liar diffrial quaio wih variabl offiis. Howvr, w shall firs sud h diffrial quaios wih osa offiis i.. quaios of h p a d a d a, a,, a d d a a g( ) d d whr ar ral osas. This quaio is o-homogous diffrial quaio ad If g ( ) h h diffrial quaio boms a d d d a a a d d whih is kow as h assoiad homogous diffrial quaio.. Iiial -Valu Problm For a liar h-ordr diffrial quaio, h problm: /,,, d d d Solv: a ( ) a ( ) a( ) a ( ) g( ) d d d Subj o: ), ( d ( ),... ( ) / / big arbirar osas, is alld a iiial-valu problm (IVP). / / Th spifid valus ( ), ( ),, ( ) ar alld iiialodiios. For h iiial-valu problm rdus o d d Solv: a ( ) a( ) a ( ) g( ) d d Coprigh Virual Uivrsi of Pakisa 9

106 Diffrial Equaios (MTH) Subj o: ),,.. Soluio of IVP ( / ) ( A fuio saisfig h diffrial quaio o I whos graph passs hrough, ) suh ha h slop of h urv a h poi is h umbr iiial valu problm.. Thorm ( Eis ad Uiquss of Soluios) / / ( is alld soluio of h L a ( ), a( ),..., a( ), a ( ) ad g () b oiuous o a irval I ad l a ( ), I. If I, h a soluio () of h iiial-valu problm is o I ad is uiqu. Eampl Cosidr h fuio This is a soluio o h followig iiial valu problm Si ad //, ( ), () d d d d / Furhr ( ) ad ( ) 6 H is a soluio of h iiial valu problm. W obsrv ha Th quaio is liar diffrial quaio. Th offiis big osa ar oiuous. Th fuio g( ) big polomial is oiuous. Th ladig offii a ( ) for all valus of. H h fuio Eampl Cosidr h iiial-valu problm is h uiqu soluio. /// // / 5 7, / ( ), (), () Clarl h problm posssss h rivial soluio. // Coprigh Virual Uivrsi of Pakisa 95

107 Diffrial Equaios (MTH) Si Th quaio is homogous liar diffrial quaio. Th offiis of h quaio ar osas. Big osa h offii ar oiuous. Th ladig offii a. H is h ol soluio of h iiial valu problm. No: If a? If a ( ) i h diffrial quaio a g( d d d ( ) a ( ) a( ) a ( ) d d d for som I h Soluio of iiial-valu problm ma o b uiqu. Soluio of iiial-valu problm ma o v is. Eampl Cosidr h fuio ad h iiial-valu problm // / 6 / ( ), () Th ad // / Thrfor () ( ) ( ) 6 6. Also ( ) () / ad () () So ha for a hoi of, h fuio ' ' saisfis h diffrial quaio ad h iiial odiios. H h soluio of h iiial valu problm is o uiqu. No ha Th quaio is liar diffrial quaio. Th offiis big polomials ar oiuous vrwhr. Th fuio g() big osa is osa vrwhr. ) Coprigh Virual Uivrsi of Pakisa 96

108 Diffrial Equaios (MTH) Th ladig offii a ( ) a (, ). H a ( ) brough o-uiquss i h soluio. Boudar-valu problm (BVP) For a d ordr liar diffrial quaio, h problm d d Solv: a ( ) a( ) a ( ) g( ) d d Subj o: ( a), ( b) is alld a boudar-valu problm. Th spifid valus ( a), ad ( b) ar alld boudar odiios... Soluio of BVP A soluio of h boudar valu problm is a fuio saisfig h diffrial quaio o som irval I, oaiig a ad b, whos graph passs hrough wo pois ( a, ) ad b, ). ( Eampl 5 Cosidr h fuio 6 W a prov ha his fuio is a soluio of h boudar-valu problm // / 6, ( ), ( ) d d Si 6 6, 6 d d d d Thrfor d d Also ( ) 6, () Thrfor, h fuio ' ' saisfis boh h diffrial quaio ad h boudar odiios. H is a soluio of h boudar valu problm.. Coprigh Virual Uivrsi of Pakisa 97

109 Diffrial Equaios (MTH).. Possibl Boudar Codiios For a d ordr liar o-homogous diffrial quaio d d a ( ) a( ) a ( ) g( ) d d all h possibl pairs of boudar odiios ar a), b), ( ( / / ( a), b ), ( a), / ( b) /, ( / ( a) /, / / ( b) / whr,, ad I Gral: / do h arbirar osas. All h four pairs of odiios miod abov ar jus spial ass of h gral boudar odiios α α ( a) ( b) β β / ( a) / ( b) whr α α, β, β {,} No ha A boudar valu problm ma hav Svral soluios. A uiqu soluio, or No soluio a all. Eampl Cosidr h fuio, os si ad h boudar valu problm Th / // // // // 6, (), ( π / ) 6( 6 6 si os os si ) γ γ Coprigh Virual Uivrsi of Pakisa 98

110 Diffrial Equaios (MTH) Thrfor, h fuio os si saisfis h diffrial quaio // 6. Now appl h boudar odiios Applig ( ) W obai So ha os si si. Bu wh w appl h d odiio ( π / ), w hav si π Sisi π, h odiio is saisfid for a hoi of, soluio of h problm is h o-paramr famil of fuios si H, hr ar a ifii umbr of soluios of h boudar valu problm. Eampl // π Solv h boudar valu problm 6, ( ),, 8 Soluio: As vrifid i h prvious ampl ha h fuio os si saisfis h diffrial quaio // 6 W ow appl h boudar odiios ( ) ad ( π / 8) So ha H Coprigh Virual Uivrsi of Pakisa 99

111 Diffrial Equaios (MTH) is h ol soluio of h boudar-valu problm. Eampl // Solv h diffrial quaio 6 subj o h boudar odiios ( ), ( π / ). Soluio:As vrifid i a arlir ampl ha h fuio saisfis h diffrial quaio 6. W ow appl h boudar odiios Thrfor So ha ( ) si Howvr ( π / ) si π or. // os si This is a lar oradiio. Thrfor, h boudar valu problm has o soluio..5 Liar Dpd A s of fuios { f ), f ( ),, f ( )} ( is said o b liarl dpd o a irval I if osas suh ha ( f ) f ( )..6 Liar Idpd A s of fuios { f ), f ( ),, f ( )} f ( ), I,,, o all zro, ( is said o b liarl idpd o a irval I if f( ) f ( ) f ( ), I,ol wh..6. Cas of wo fuios If h h s of fuios boms{ f ), f ( )} If w suppos ha f ) f ( ) ( ( Also ha h fuios ar liarl dpd o a irval I h ihr or. Coprigh Virual Uivrsi of Pakisa

112 Diffrial Equaios (MTH) L us assum ha, h ( ) f ( ) f.h f ( ) is h osa mulipl of f ( ).Covrsl, if w suppos f ) f ( ) Th ) f ( ) f ( ), I ( ( So ha h fuios ar liarl dpd baus. H, w olud ha: A wo fuios f( ) ad f ( ) ar liarl dpd o a irval I if ad ol if o is h osa mulipl of h ohr. A wo fuios ar liarl idpd wh ihr is a osa mulipl of h ohr o a irval I. I gral a s of fuios { f ), f ( ),, f ( )} Eampl Th fuios If w hoos ( is liarl dpd if a las o of hm a b prssd as a liar ombiaio of h rmaiig. f ( ) si, (, ) f ( ) si os, (, ) ad h si si os ( si os ) si os H, h wo fuios f ( ) ad f ( ) ar liarl dpd. Eampl Cosidr h fuios π f( ) os, f ( ) si, ( π /, / ), π f( ) s, f ( ) a, ( π /, / ) If w hoos,,, h f ( ) f os os si ( ) si f a ( ) s f a Thrfor, h giv fuios ar liarl dpd. No ha ( ) a Coprigh Virual Uivrsi of Pakisa

113 Diffrial Equaios (MTH) Th fuio f ( ) a b wri as a liar ombiaio of ohr hr fuios ( ) f ( ), f ad f ( ) baus s os si a. Eampl Cosidr h fuios Th f ( ), (, ) f f mas ha ( ), ( ), (, ) (, ) f( ) f ( ) f( ) ( ) or ( ) Equaig offiis of ad Thrfor osa rms w obai H, h hr fuios f ), f ( ) ad f ( ) ar liarl idpd..7 Wroskia ( Suppos ha h fuio f( ), f ( ),, f ( ) posssss a las drivaivs h h drmia / f f f f f / f is alld Wroskia of h fuios f( ), f ( ),, f ( ) ad is dod b W ( f ), f ( ),, f ( )). ( f f f /.8 Thorm (Cririo for Liarl Idpd Fuios) Suppos h fuios f( ), f ( ),, f ( ) possss a las - drivaivs o a irval I. If Coprigh Virual Uivrsi of Pakisa

114 Diffrial Equaios (MTH) W f( ), f ( ),, f ( )) ( for a las o poi i I, h fuios f( ), f ( ),, f ( ) ar liarl idpd o h irval I. No ha his is ol a suffii odiio for liar idpd of a s of fuios. I ohr words: If f ), f ( ),, f ( ) posssss a las drivaivs o a irval ad ar ( liarl dpd o I, h W f ( ), f ( ),, f ( )), I ( Howvr, h ovrs is o ru. i.. a Vaishig Wroskia dos o guara liar dpd of fuios. Eampl Th fuios f f ( ) si ( ) os ar liarl dpd baus si ( os ) W obsrv ha for all (, ) W ( f ( ), f ( ) ) si si os os si Eampl Cosidr h fuios f m si si si os si os os si [si si [si si [si os ] os os ( ), f ( ), m m Th fuios ar liarl idpd baus f( ) f ( ) if ad ol if as m m m ] si ] Coprigh Virual Uivrsi of Pakisa

115 Diffrial Equaios (MTH) Now for all R m m (, ) W m m m m m ( ) ( m ) m m m m Thus f ad f ar liarl idpd of a irval o -ais. Eampl If α ad β ar ral umbrs, β, h h fuios α α os β ad si β ar liarl idpd o a irval of h -ais baus α α ( os β, si β) W β α α os β si β α α os β β α α si β os β α α si β Eampl Th fuios β f α α ( os β si β) β. ( ), f ( ), ad f ( ) ar liarl idpd o a irval of h -ais baus for all R, w hav (,, ) W.9 Eris. Giv ha is a wo-paramr famil of soluios of h diffrial quaio Coprigh Virual Uivrsi of Pakisa

116 Diffrial Equaios (MTH) o ( ). Giv ha,, fid a mmbr of h famil saisfig h boudar odiios ( ), ( ). os si is a hr-paramr famil of soluios of h diffrial quaio o h irval ( ) odiios ( π ), ( π ), ( π ). Giv ha,, fid a mmbr of h famil saisfig h iiial. l is a wo-paramr famil of soluios of h diffrial quaio o (, ). Fid a mmbr of h famil saisfig h iiial odiios ( ), ( ). Drmi whhr h fuios i problms -7 ar liarl idpd or,. dpd o ( ). ( ) ( ) ( ) f f, f, 5. ( ) ( ) ( ) f, f, f f os, f, f os, 6. ( ) ( ) ( ) 7. f ( ) f ( ), f ( ) sih Show b ompuig h Wroskia ha h giv fuios ar liarl idpd o h idiad irval. 8. a, o ; (-, ) 9. -,, ; (, )., l, l ; (, ) Coprigh Virual Uivrsi of Pakisa 5

117 Diffrial Equaios (MTH) Soluios of Highr Ordr Liar Equaios. Prlimiar Thor I ordr o solv a h ordr o-homogous liar diffrial quaio d d d a ( ) a ( ) a ( ) a ( ) g( ) d d d w firs solv h assoiad homogous diffrial quaio d d d a ( ) a ( ) a( ) a ( ) d d d Thrfor, w firs ora upo h prlimiar hor ad h mhods of solvig h homogous liar diffrial quaio. W rall ha a fuio f () ha saisfis h assoiad homogous quaio a d d d d d d ( ) a ( ) a ( ) a ( ) is alld soluio of h diffrial quaio.. Suprposiio Priipl,,, Suppos ha ar soluios o a irval I of h homogous liar diffrial quaio Th a d d d d d d ( ) a ( ) a ( ) a ( ) ( ) ( ) ( ),,,, big arbirar osas is also a soluio of h diffrial quaio. No ha A osa mulipl ( ) of a soluio ( ) of h homogous liar diffrial quaio is also a soluio of h quaio. Th homogous liar diffrial quaios alwas possss h rivial soluio. Th suprposiio priipl is a propr of liar diffrial quaios ad i dos o hold i as of o-liar diffrial quaios. Eampl Th fuios diffrial quaio,, ad all saisf h homogous Coprigh Virual Uivrsi of Pakisa 6

118 Diffrial Equaios (MTH) o ( ) d d d d 6 d d 6,. Thus, ad ar all soluios of h diffrial quaio Now suppos ha Th Thrfor Thus d d d d d d 6 d d d 8 d 6 d. 9 7 d ( 6 6 ) ( 8 6 ) ( ).. ( ) ( ) ( 6 6) is also a soluio of h diffrial quaio. Eampl Th fuio o (, ). Now osidr ad So ha 6 H h fuio Th Wroskia.. is a soluio of h homogous liar quaio is also a soluio of h giv diffrial quaio. Suppos ha, ar soluios, o a irval I, of h sod ordr homogous liar diffrial quaio Coprigh Virual Uivrsi of Pakisa 7

119 Diffrial Equaios (MTH) a d d a a Th ihr W ( ), I d, or W ( ), I, To vrif his w wri h quaio as Now W ( ) d d Pd Q d d, Diffriaig w.r.o, w hav dw d Si ad ar soluios of h diffrial quaio Thrfor d Pd Q d d Q P Q P Muliplig s quaio b ad d b h hav P Q P Q Subraig h wo quaios w hav: ( ) P( ) dw or PW d This is a liar s ordr diffrial quaio i W, whos soluio is Thrfor If If Pd W h ( ) I W,, h W ( ), I, Coprigh Virual Uivrsi of Pakisa 8

120 Diffrial Equaios (MTH) H Wroskia of ad is ihr idiall zro or is vr zro o I. I gral If,,, ar soluios, o a irval I, of h homogous h ordr liar diffrial quaio wih osas offiis Th a d d d a a a d Eihr W (,,, ), I d or W (,,, ), I. Liar Idpd of Soluios Suppos ha,,, ar soluios, o a irval I, of h homogous liar h-ordr diffrial quaio d d d a d d d Th h s of soluios is liarl idpd o I if ad ol if W, I ohr words Th soluios ( ) a ( ) a ( ) a ( ),,, ( ),, ar liarl dpd if ad ol if (, ), I W,,. Fudamal S of Soluios A s {,,, } of liarl idpd soluios, o irval I, of h homogous liar h-ordr diffrial quaio a d d d d ( ) a ( ) a ( ) a ( ) d d d Coprigh Virual Uivrsi of Pakisa 9

121 Diffrial Equaios (MTH) is said o b a fudamal s of soluios o h irval I... Eis of a Fudamal S Thr alwas iss a fudamal s of soluios for a liar h-ordr homogous diffrial quaio o a irval I. a d d d d d d ( ) a ( ) a ( ) a ( ).5 Gral Soluio-Homogous Equaios Suppos ha {,,, } is a fudamal s of soluios, o a irval I, of h homogous liar h-ordr diffrial quaio d d d ( ) a ( ) a ( ) a ( ) a d d d Th h gral soluio of h quaio o h irval I is dfid o b Hr,, ar arbirar osas., ( ) ( ) ( ) Eampl Th fuios ad ar soluios of h diffrial quaio 9 Si W, 6, I Thrfor ad from a fudamal s of soluios o ( ) soluio of h diffrial quaio o h (, ) is Eampl Cosidr h fuio sih 5 Th osh 5, 9 sih 5 or 9, Thrfor 9,. H gral 6sih 5 Coprigh Virual Uivrsi of Pakisa

122 Diffrial Equaios (MTH) H sih 5 is a pariular soluio of diffrial quaio. 9 Th gral soluio of h diffrial quaio is Choosig, 7 W obai sih 5 H, h pariular soluio has b obaid from h gral soluio. Eampl d d d Cosidr h diffrial quaio 6 6 d d d ad suppos ha ad, Th d d d d d d Thrfor d d d 6 6 d d d 6 6 d or d 6 d 6 d d d Thus h fuio is a soluio of h diffrial quaio. Similarl, w a vrif ha h ohr wo fuios i.. ad also saisf h diffrial quaio. Now for all R Coprigh Virual Uivrsi of Pakisa

123 Diffrial Equaios (MTH) W,, 9 6 I Thrfor,, ad form a fudamal soluio of h diffrial quaio o ( ),. W olud ha is h gral soluio of h diffrial quaio o h irval ( ).6 No-Homogous Equaios A fuio,. p ha saisfis h o-homogous diffrial quaio d d d a d d d ( ) a ( ) a ( ) a ( ) g( ) ad is fr of paramrs is alld h pariular soluio of h diffrial quaio Eampl Suppos ha So ha p 9 p 9 7 p ( ) p Thrfor is a pariular soluio of h diffrial quaio 9 7 p p p Eampl Suppos ha p, p p 6 p p 8 p 6 8 Thrfor ( ) 6 Thrfor p is a pariular soluio of h diffrial quaio Complmar Fuio Th gral soluio of h homogous liar diffrial quaio d d d a a a a d d d ( ) ( ) ( ) ( ) is kow as h omplmar fuio for h o-homogous liar diffrial quaio. Coprigh Virual Uivrsi of Pakisa

124 Diffrial Equaios (MTH) a d d d d ( ) a ( ) a ( ) a ( ) g( ) d d.8 Gral Soluio of No-Homogous Equaios Suppos ha Th pariular soluio of h o-homogous quaio is p. a d d d d ( ) a ( ) a ( ) a ( ) g( ) d d Th omplmar fuio of h o-homogous diffrial quaio is a d d d d ( ) a ( ) a ( ) a ( ) d d. Th gral soluio of h o-homogous quaio o h irval I is giv b p or ( ) ( ) ( ) ( ) ( ) ( ) p H Gral Soluio Complmar soluio a pariular soluio. Eampl Suppos ha Th H p p, p p d p d p d p 6 6 d d d p p is a pariular soluio of h o-homogous quaio d d d 6 6 d d d Now osidr Th p Coprigh Virual Uivrsi of Pakisa

125 Diffrial Equaios (MTH) d d d d d d Si, d d d d d d 6 6 ( ) ( ) ( ) Thus is gral soluio of assoiad homogous diffrial quaio 6 6 d d d d d d H gral soluio of h o-homogous quaio is p.9 Suprposiio Priipl for No-homogous Equaios Suppos ha pk p p,,, do h pariular soluios of h k diffrial quaio ( ) ( ) ( ) ( ) ( ) ( ) ( ), i g a a a a i k,,, o a irval I. Th ( ) ( ) ) ( p p p p k is a pariular soluio of ( ) ( ) ( ) ( ) ( ) ( ) ( ) k g g g a a a a Eampl Cosidr h diffrial quaio 8 6 Suppos ha p p p,, Coprigh Virual Uivrsi of Pakisa

126 Diffrial Equaios (MTH) Th p 8 6 p p Thrfor p is a pariular soluio of h o-homogous diffrial quaio 6 8 Similarl, i a b vrifid ha ad p ar pariular soluios of h quaios: ad rspivl. H p - p p p p is a pariular soluio of h diffrial quaio 6 8. Eris Vrif ha h giv fuios form a fudamal s of soluios of h diffrial quaio o h idiad irval. Form h gral soluio.. ;,, (, ). 5 ; os, si, (, ). ; os( l ), si( l ), (, ) / /. ;,, (, ) 5. 6 ;, (, ) 6. ; osh, sih, (, ) Coprigh Virual Uivrsi of Pakisa 5

127 Diffrial Equaios (MTH) Vrif ha h giv wo-paramr famil of fuios is h gral soluio of h ohomogous diffrial quaio o h idiad irval. 7. s, si si ( os ) l( os ), ( π /, π / ) os 8., , 6, (, ).. / 5,, 5 6 (, ) Coprigh Virual Uivrsi of Pakisa 6

128 Diffrial Equaios (MTH) 5 Cosruio of a Sod Soluio 5. Gral Cas Cosidr h diffrial quaio d d a ( ) a( ) a ( ) d d W divid b a ( ) o pu h abov quaio i h form // / P( ) Q( ) Whr P () ad Q () ar oiuous o som irval I. Suppos ha ( ), I is a soluio of h diffrial quaio // / Th P Q W dfi u ( ) ( ) h / / u u, u u u This implis ha w mus hav // // / / // P Q u[ P Q] u ( P) u zro / / If w suppos w u, h / // / u ( P) u / / w ( P) w Th quaio is sparabl. Sparaig variabls w hav from h las quaio Igraig / dw. ( P) d w l l Pd w l w Pd Pd w Pd d w Coprigh Virual Uivrsi of Pakisa 7

129 Diffrial Equaios (MTH) or u / Pd Igraig agai, w obai Pd u d Pd H u( ) ( ) ( ) d ( ). Choosig ad, w obai a sod soluio of h diffrial quaio Th Woolski Pd ( ) d W ( ( ), ( ) ) Pd d Pd Pd d Pd, Thrfor ( ) ad ( ) ar liar idpd s of soluios. So ha h form a fudamal s of soluios of h diffrial quaio // / P( ) Q( ) H h gral soluio of h diffrial quaio is ( ) ( ) ( ) Coprigh Virual Uivrsi of Pakisa 8

130 Diffrial Equaios (MTH) Eampl Giv ha is a soluio of // / Fid gral soluio of h diffrial quaio o h irval (, ). Soluio: Th quaio a b wri as Th d soluio is giv b // /, or ( ) Pd d d l d d d l H h gral soluio of h diffrial quaio o (, ) is giv b or Eampl Vrif ha is a soluio of si / ( ) // l o (,π ). Fid a sod soluio of h quaio. Coprigh Virual Uivrsi of Pakisa 9

131 Diffrial Equaios (MTH) Soluio: Th diffrial quaio a b wri as // / ( ) Th d soluio is giv b si Thrfor si Pd d d d si ( ) si si s d d si os ( o ) Thus h sod soluio is os H, gral soluio of h diffrial quaio is 5. Ordr Rduio Eampl Giv ha si is a soluio of h diffrial quaio Fid sod soluio of h quaio Soluio // os 6, Coprigh Virual Uivrsi of Pakisa

132 Diffrial Equaios (MTH) W wri h giv quaio as: // 6 6 So ha P( ) Thrfor Pd d Thrfor, usig h formula 6 d d Pd W our a igral ha is diffiul or impossibl o valua. d H, w olud somims us of h formula o fid a sod soluio is o suiabl. W d o r somhig ls. Alraivl, w a r h rduio of ordr o fid. For his purpos, w agai dfi h Subsiuig h valus of ( ) u ( ) ( ) or u( ). u u / u 6 u 6u, i h giv diffrial quaio w hav 6 Coprigh Virual Uivrsi of Pakisa

133 Diffrial Equaios (MTH) ( u 6 u 6u) 6u 5 6 or u u 6 or u u, If w ak w u h w / 6 w This is sparabl as wll as liar firs ordr diffrial quaio i w. For usig h lar, w fid h igraig faor 6 d 6 l 6 I. F 6 Muliplig wih h IF, w obai 6 w 6 5 w d or ( 6 w) d Igraig w.r.., w hav 6 w / or u 6 Igraig o agai, givs Thrfor Choosig ad 5, w obai Thus h sod soluio is giv b u 5 5 u 5 H, gral soluio of h giv diffrial quaio is Coprigh Virual Uivrsi of Pakisa

134 Diffrial Equaios (MTH) i.. ( ) Whr ad ar osas. 5. Eris / Fid h d soluio of ah of Diffrial quaios b rduig ordr or b usig h formula. // /. ; // /. ; //. ; si 9 // 5. 5 ; // / 5. 6 ; // / 6. 6 ; // 7. ; l // / 8. ) ; ( // / 9. 5 ; os(l ) // /. ( ) ; Coprigh Virual Uivrsi of Pakisa

135 Diffrial Equaios (MTH) 6 Homogous Liar Equaios wih Cosa Coffiis d W kow ha h liar firs ordr diffrial quaio m d m, as. m big a osa, has h poial soluio o ( ) Th qusio? Th qusio is whhr or o h poial soluios of h highr-ordr diffrial quaios ( ) ( ) // / a a a a a,,. is o ( ) I fa all h soluios of his quaio ar poial fuios or osrud ou of poial fuios. Rall ha h liar diffrial of ordr is a quaio of h form a g( d d d ( ) a ( ) a( ) a ( ) d d d 6. Mhod of Soluio d d Takig, h h-ordr diffrial quaio boms a a a d d d d This quaio a b wri as a b d d W ow r a soluio of h poial form m ) m m ad Subsiuig i h diffrial quaio, w hav m ( am bm ) m, hrfor am bm Si, (, ) m m This algbrai quaio is kow as h Auiliar quaio (AE).Th soluio of h auiliar quaio drmis h soluios of h diffrial quaio. 6.. Cas (Disi Ral Roos) If h auiliar quaio has disi ral roos m ad m h w hav h followig wo m m soluios of h diffrial quaio. ad Ths soluios ar liarl idpd baus ( m m ) W / ( m m ) (, ) / Si m m ad ( m ) m, hrfor W (, ) (, ). Coprigh Virual Uivrsi of Pakisa

136 Diffrial Equaios (MTH) H ad form a fudamal s of soluios of h diffrial quaio. Th gral soluio of h diffrial quaio o ( ) m m 6.. Cas (Rpad Roos), is If h auiliar quaio has ral ad qual roos i.. m m, m wih m m Th w obai ol o poial soluio m b To osru a sod soluio w rwri h quaio i h form a a Comparig wih P Q W mak h idifiaio b P a Thus a sod soluio is giv b b Pd a m d d m Si h auiliar quaio is a quadrai algbrai quaio ad has qual roos Thrfor, Dis. b a W kow from h quadrai formula w hav b m.thrfor a H h gral soluio is 6.. Cas (Compl Roos) b ± m m b a a m d m m m m m ( ) If h auiliar quaio has ompl roos α ± iβ h, wih m α iβ ad m α iβ, whr α > ad β > ar ral, h gral soluio of h diffrial ( α iβ ) ( α iβ ) quaio is Firs w hoos h followig wo pairs of valus of ad,, ha θ,h w hav i osθ i siθ, θ R. ( α iβ ) ( α iβ ) ( α iβ ).W kow b h Eulr s Formula ( α iβ ) Coprigh Virual Uivrsi of Pakisa 5

137 Diffrial Equaios (MTH) Usig his formula, w a simplif h soluios ad as α iβ iβ α ( ) os β α iβ iβ α ( ) i si β W a drop osa o wri. α os β, α si β α α Th Wroskia: W ( os β, si β) β α α α Thrfor, os( β ), si( β ) form a fudamal s of soluios of h,.h gral soluio of h diffrial quaio is diffrial quaio o ( ) α α os β si β ( os β si β) Eampl: Solv 5 Soluio: Th giv diffrial quaio is 5 Pu m m m, w hav ( 5m ) m m α. Subsiuig i h giv diffrial quaio, m m. Si m, h auiliar quaio is m 5m as m m m, m ( )( ) Thrfor, h auiliar quaio has disi ral roos m ad m H h gral soluio of h diffrial quaio is Eampl Solv 5 ( / ) Soluio: W pu m m m m, m m Subsiuig i h giv diffrial quaio, w hav ( m m 5) Si m, h auiliar quaio is m m 5 ( 5) m m 5, 5.Thus h auiliar quaio has rpad ral roos i. m 5 m. H gral soluio of h diffrial quaio is ( ) Eampl Solv h iiial valu problm: -, ( ) ( ) Soluio: Giv ha h diffrial quaio Pu m m m, m m Coprigh Virual Uivrsi of Pakisa 6

138 Diffrial Equaios (MTH) m Subsiuig i h giv diffrial quaio, w hav: ( m m ) Si m, h auiliar quaio is m m B quadrai formula, h soluio of h auiliar quaio is Thus h auiliar quaio has ompl roos m i, m i ± 6 5 m ± i H gral soluio of h diffrial quaio is ( si ) os Eampl Solv h diffrial quaios (a) k, (b) k Soluio Firs osidr h diffrial quaio k, Pu m m m m ad m. Subsiuig i h giv diffrial quaio, w hav: ( m k ) Si, h auiliar quaio is m k m ±ki, Thrfor, h auiliar quaio has ompl roos m ki, m ki H gral soluio of h diffrial quaio is os k si k m d N osidr h diffrial quaio k d m Subsiuig valus ad, m k m w hav. ( ) Si, h auiliar quaio is m k m ± k Thus h auiliar quaio has disi ral roos m k, m k k H h gral soluio is. 6. Highr Ordr Equaios If w osidr h ordr homogous liar diffrial quaio k d d d a a a a d d d Th, h auiliar quaio is a h dgr polomial quaio am a m am a 6.. Cas (Ral disi roos) If h roos m, m,, m of h auiliar quaio ar all ral ad disi, h h m m m gral soluio of h quaio is m Coprigh Virual Uivrsi of Pakisa 7

139 Diffrial Equaios (MTH) 6.. Cas (Ral & rpad roos) W suppos ha m is a roo of muliplii of h auiliar quaio, h i a b show ha m m,,, m ar liarl idpd soluios of h diffrial quaio. H gral soluio of h diffrial quaio is m m m 6.. Cas (Compl roos) Suppos ha offiis of h auiliar quaio ar ral. W fi a 6, all roos of h auiliar ar ompl, aml α± iβ, α ± iβ, α± iβ Th h gral soluio of h diffrial quaio α ( os β si β ) ( os β si β ) α α ( 5os β 6si β) If 6, wo roos of h auiliar quaio ar ral ad qual ad h rmaiig ar ompl, aml α ± i β, α ± iβ Th h gral soluio is α m m α ( os β si β ) ( os β si β ) 5 6 If m α iβ is a ompl roo of muliplii k of h auiliar quaio. Th is ojuga m α iβ is also a roo of muliplii k. Thus from Cas, h diffrial quaio has k soluios ( α iβ ) ( α iβ ) ( α iβ ) k ( α iβ ),,,, ( α iβ ) ( α iβ ) ( α iβ ) k ( α iβ ),,,, B usig h Eulr s formula, w olud ha h gral soluio of h diffrial quaio is a liar ombiaio of h liarl idpd soluios α α α k α os β, os β, os β,, os β α α α k α si β, si β, si β,, si β Thus if k h α [ os β d d d si β 6. Solvig h Auiliar Equaio ( ) ( ) ] Rall ha h auiliar quaio of h dgr diffrial quaio is h dgr polomial quaio Solvig h auiliar quaio ould b diffiul ( m), > P Coprigh Virual Uivrsi of Pakisa 8

140 Diffrial Equaios (MTH) O wa o solv his polomial quaio is o guss a roo m. Th m m is a faor of h polomial P (m). Dividig wih m m shiall or ohrwis, w fid h faorizaio P ( m) ( m m ) Q( m) W h r o fid roos of h quoi i.. roos of h polomial quaio Q ( m) p No ha if m is a raioal ral roo of h quaio q P ( m), > h p is a faor of a ad q of a. B usig his fa w a osru a lis of all possibl raioal roos of h auiliar quaio ad s ah of hm b shi divisio. Eampl Solv h diffrial quaio Soluio:Giv h diffrial quaio m. Pu / m // m /// m m, m ad m Subsiuig his i h giv diffrial quaio, w hav m ( m m ) m Si m m So ha h auiliar quaio is m m Soluio of h AE If w ak m h w s ha m m Thrfor m saisfis h auiliar quaios so ha m- is a faor of h polomial m m. B shi divisio, w a wri m m ( m )( m m ) So, m m ( m )( m ) ( m )( m ) m,, H soluio of h diffrial quaio is Eampl /// // / Solv 5 /// // / Soluio: Giv h diffrial quaio 5 Pu m / m // m /// m m, m ad m Coprigh Virual Uivrsi of Pakisa 9

141 Diffrial Equaios (MTH) Thrfor h auiliar quaio is m 5m m Soluio of h auiliar quaio: a) a ad all is faors ar: p : ±, ±, ± b) a ad all is faors ar: q : ±, ± ) Lis of possibl raioal roos of h auiliar quaio is p : -,, -,, -,,,,,,, q d) Tsig ah of hs sussivl b shi divisio w fid 5 6 Cosqul a roo of h auiliar quaio is m Th offiis of h quoi ar 6 Thus w a wri h auiliar quaio as: ( m )( m 6m ) m or m 6 m m or m ± i (/ ) H soluio of h giv DE is: ( si ) os d d Eampl Solv h diffrial quaio d d d d Soluio: Giv h diffrial quaio. d d Pu m m m m, m m Subsiuig i h diffrial quaio, w obai ( m m ) m Si, h auiliar quaio is m m ( m ) m ± i, ± i m m i ad m m i Thus i is a roo of h auiliar quaio of muliplii ad so is i. Now α ad β.h h gral soluio of h diffrial quaio is [ )os ( d d )si ] ( os d si os dsi Coprigh Virual Uivrsi of Pakisa

142 Diffrial Equaios (MTH) Eris Fid h gral soluio of h giv diffrial quaios.. // 8. // /. // /. // / 5. /// // / 6. /// 5 // 7. /// // / Solv h giv diffrial quaios subj o h idiad iiial odiios. 8. /// // / 5 6, / // () (), () 9. d / // ///, (), (), (), () 5 d d. / // ///, () () (), () d Coprigh Virual Uivrsi of Pakisa

143 Diffrial Equaios (MTH) 7 Mhod of Udrmid Coffiis(Suprposiio Approah) Rall. Tha a o-homogous liar diffrial quaio of ordr is a quaio of h form a d d a d d a a g( ) d d Th offiis a, a,, a a b fuios of. Howvr, w will disuss quaios wih osa offiis.. Tha o obai h gral soluio of a o-homogous liar diffrial quaio w mus fid: Th omplmar fuio, whih is gral soluio of h assoiad homogous diffrial quaio. A pariular soluio of h o-homogous diffrial quaio. p. Tha h gral soluio of h o-homogous liar diffrial quaio is giv b Fidig Gral soluio Complmar fuio Pariular Igral Complmar fuio has b disussd i h prvious lur. I h hr lurs w will disuss mhods for fidig a pariular igral for h ohomogous quaio, aml Th mhod of udrmid offiis-suprposiio approah Th mhod udrmid offiis-aihilaor opraor approah. Th mhod of variaio of paramrs. Th Mhod of Udrmid Coffii Th mhod of udrmid offiis dvlopd hr is limid o o-homogous liar diffrial quaios Tha hav osa offiis, ad Whr h fuio g () has a spifi form. 7. Th form of Ipu fuio g () Th ipu fuio g () a hav o of h followig forms: A osa fuio k. A polomial fuio A poial fuio Th rigoomri fuios si( β ), os( β ) Fii sums ad produs of hs fuios. Coprigh Virual Uivrsi of Pakisa

144 Diffrial Equaios (MTH) Ohrwis, w ao appl h mhod of udrmid offiis. 7. Soluio Sps Cosis of prformig h followig sps. Sp Drmi h form of h ipu fuio g (). Sp Sp Sp Sp 5 Sp 6 Assum h gral form of aordig o h form of g () p Subsiu i h giv o-homogous diffrial quaio. Simplif ad qua offiis of lik rms from boh sids. Solv h rsulig quaios o fid h ukow offiis. Subsiu h alulad valus of offiis i assumd 7.. Rsriio o Ipu fuio g Th ipu fuio g is rsrid o hav o of h abov sad forms baus of h raso: Th drivaivs of sums ad produs of polomials, poials ar agai sums ad produs of similar kid of fuios. // / Th prssio a b has o b idiall qual o h ipu p p p fuio g (). Thrfor, o mak a duad guss, Cauio! p p is assurd o hav h sam form as g. I addiio o h form of h ipu fuio g (), h duad guss for p mus ak io osidraio h fuios ha mak up h omplmar fuio. No fuio i h assumd diffrial quaio. This mas ha h assumd ha duplia rms i. mus b a soluio of h assoiad homogous p p should o oai rms Takig for grad ha o fuio i h assumd p is dupliad b a fuio i, som forms of g ad h orrspodig forms of p ar giv i h followig abl. Coprigh Virual Uivrsi of Pakisa

145 Diffrial Equaios (MTH) 7. Trial pariular soluios Numbr Th ipu fuio g () Th assumd pariular soluio A osa.g. A 5 7 A B A B A B C D p 5 si A os B si 6 os A os B si ( 9 ) 5 A 5 ( A B) ( A B C) 5 si A os B si si ( A B C )os ( A B C )si 5 os ( A B) os ( C D) si 7. Ipu fuio g( ) as a sum Suppos ha Th ipu fuio g( ) abov abl i.. g ( ) g ( ) g ( ) gm ( ). Th rial forms orrspodig o g ( ) g ( ),, g ( ) osiss of a sum of m rms of h kid lisd i h, m b,,,. p p p m Coprigh Virual Uivrsi of Pakisa

146 Diffrial Equaios (MTH) Th h pariular soluio of h giv o-homogous diffrial quaio is p p p I ohr words, h form of p is a liar ombiaio of all h liarl idpd fuios grad b rpad diffriaio of h ipu fuio g (). // / Eampl Solv 6 Soluio: Complmar fuio: To fid, w firs solv h assoiad homogous p m quaio // / W pu m, m m, m m Th h assoiad homogous quaio givs ( m m ) m Thrfor, h auiliar quaio is m m as m, Usig h quadrai formula, roos of h auiliar quaio ar m ± 6 Thus w hav ral ad disi roos of h auiliar quaio m 6 ad m 6. H h omplmar fuio is ( 6) ( 6) N w fid a pariular soluio of h o-homogous diffrial quaio. Pariular Igral Si h ipu fuio g ( ) 6 is a quadrai polomial. Thrfor, w assum ha p A B C / p A B ad p A // // / p p A 8A B A B C p Subsiuig i h giv quaio, w hav A 8A B A B C 6 Or A (8A B) (A B C) 6 Equaig h offiis of h lik powrs of, w hav - A, 8A - B -, A B - C 6 Solvig his ssm of quaios lads o h valus A, B 5, C 9. Thus a pariular soluio of h giv quaio is Coprigh Virual Uivrsi of Pakisa 5

147 Diffrial Equaios (MTH) 5 p 9.H, h gral soluio of h giv o-homogous diffrial quaio is giv b p 5 9 ( 6) ( Eampl Solv h diffrial quaio 6) // / si Soluio: Complmar fuio: To fid, w solv h assoiad homogous // / diffrial quaio. Pu m m m, m m.subsiu i h giv diffrial quaio o obai h auiliar quaio m ± i m m H, h auiliar quaio has ompl roos. H h omplmar fuio is (/ ) os si Pariular Igral Si sussiv diffriaio of g( ) si produ si ad os.thrfor, w ilud boh of hs rms i h assumd pariular soluio, s abl Aos Bsi. Asi B os. 9Aos 9Bsi. p p p // p / p ( 8A B)os (A 8B)si. p Subsiuig i h giv diffrial quaio: ( 8A B)os (A 8B)si os si. From h rsulig quaios 8 A B, A 8B Solvig hs quaios, w obai A 6 / 7, B 6 / 7 A pariular soluio of h quaio is 6 6 os si p 7 7 H h gral soluio of h giv o-homogous diffrial quaio is Eampl Solv (/ ) 6 6 os si os si 7 7 // / 5 6 Soluio: Complmar fuio Coprigh Virual Uivrsi of Pakisa 6

148 Diffrial Equaios (MTH) To fid, w solv h assoiad homogous quaio // / Pu m m m, m m Subsiu i h giv diffrial quaio o obai h auiliar quaio m m m, ( m )( m ) Thrfor, h auiliar quaio has ral disi roo m, m Thus h omplmar fuio is Pariular igral Si g ) ( 5) 6 g ( ) g ( ) ( Corrspodig o g ( ) : A B Corrspodig o g ( ) : p ( C D) p. Th suprposiio priipl suggss ha w assum a pariular soluio p p p A B C D p ( ) C D C p ( ).Subsiuig i h giv: // / p p p Simplifig ad groupig lik rms C D C A C A C D C p ( ) D C A B C D // / p p p A A B C (C D) 5 6. Subsiuig i h o-homogous diffrial quaio, w hav A A B C (C D) 5 6 Now quaig osa rms ad offiis of, ad A B 5, A, C 6, C D Solvig hs algbrai quaios, w fid A, C, B 9, w obai D - Thus, a pariular soluio of h o-homogous quaio is p ( ) ( 9) ( ) Coprigh Virual Uivrsi of Pakisa 7

149 Diffrial Equaios (MTH) gral soluio: p ( ) ( 9) - ( ) 7.5 Dupliaio bw p ad If a fuio i h assumd p is also prs i h his fuio is a soluio of h assoiad homogous diffrial quaio. I his as h obvious assumpio for h form of p is o orr. I his as w suppos ha h ipu fuio is mad up of rms of kids i.. g( ) g( ) g ( ) g ( ) ad orrspodig o his ipu fuio h assumd pariular soluio p is If a p i p p p oai rms ha duplia rms i, h ha p p i mus b muliplid wih, big h las posiiv igr ha limias h dupliaio. Eampl Fid a pariular soluio of h followig o-homogous diffrial quaio // 5 / 8. Soluio: To fid, w solv h assoiad homogous diffrial quaio // 5 / W pu m i h giv quaio, so ha h auiliar quaio is m 5m m, g( ) 8 A p Subsiuig i h giv o-homogous diffrial quaio, w obai A 5 A A 8 8 Clarl w hav mad a wrog assumpio for p, as w did o rmov h dupliaio. Si A is prs i. Thrfor, i is a soluio of h assoiad homogous diffrial quaio // 5 / To avoid his w fid a pariular soluio of h form p A W oi ha hr is o dupliaio bw ad his w assumpio for / // Now p A A, p A A.Subsiuig i h giv diffrial quaio, w obai A A 5A 5A A 8. p Coprigh Virual Uivrsi of Pakisa 8

150 Diffrial Equaios (MTH) or A 8 A 8. So ha a pariular soluio of h giv quaio is giv b ) p (8 H, h gral soluio of h giv quaio is (8 / ) Eampl 5 (a) Drmi h form of h pariular soluio // 8 / // (b) os. Soluio: (a) To fid w solv h assoiad homogous diffrial quaio // 8 / 5 Pu m h auiliar quaio is m 8m 5 m ± i Roos of h auiliar quaio ar ompl ( os si ) Th ipu fuio is g( ) 5 7 (5 7) Thrfor, w assum a pariular soluio of h form A B C D p ( ) Noi ha hr is o dupliaio bw h rms i p ad h rms i. Thrfor, whil prodig furhr w a asil alula h valu A, B, C ad D. // (b) Cosidr h assoiad homogous diffrial quaio Si g( ) os.thrfor, w assum a pariular soluio of h form p ( A B)os ( C D) si.agai obsrv ha hr is o dupliaio of rms bw ad Eampl 6 p Drmi h form of a pariular soluio of 5si 7 // / 6 Soluio: To fid, w solv h assoiad homogous diffrial quaio // /.Pu m,h h auiliar quaio is ± m m m i (/ ) os si g( ) 5si 7 g ( ) g ( ) g ( ) 6 Coprigh Virual Uivrsi of Pakisa 9

151 Diffrial Equaios (MTH) Corrspodig o g ( ) : p A B C Corrspodig o g ( ) 5si : p D os E si Corrspodig o g : p ( F G) 6 ( ) 7 6 H, h assumpio for h pariular soluio is p p p p 6 p A B C D os E si ( F G) No rm i his assumpio duplia a rm i h omplmar fuio 7 Eampl 7 Fid a pariular soluio of // / Soluio: Cosidr h assoiad homogous quaio // / Pu m.th h auiliar quaio is : m m m ( m ), Roos of h auiliar quaio ar ral ad qual. Thrfor, Si g ( ).Thrfor, w assum ha p A This assumpio fails baus of dupliaio bw ad p. W mulipl wih Thrfor, w ow assum p A.Howvr, h dupliaio is sill hr. Thrfor, w agai mulipl wih ad assum p A Si hr is o dupliaio, his is apabl form of h rial // si, Eampl 8 Solv h iiial valu problm: / ( π ), ( π ) Soluio Cosidr h assoiad homogous diffrial quaio //.Pu p m Th h auiliar quaio is m m ± i Th roos of h auiliar quaio ar ompl. Thrfor, h omplmar fuio is os si Si g ( ) si g( ) g ( ) Thrfor, w assum ha p A B, p C os Dsi Coprigh Virual Uivrsi of Pakisa

152 Diffrial Equaios (MTH) So ha p A B C os Dsi Clarl, hr is dupliaio of h fuios os ad si. To rmov his dupliaio w mulipl wih. Thrfor, w assum ha So ha p p A B C os Dsi. C si C os D os Dsi p // p p A B C si D os Subsiuig io h giv o-homogous diffrial quaio, w hav A B C si D os si Equaig osa rms ad offiis of, si, os, w obai B, A, C, D So ha A, B, C 5, D Thus p 5 os H h gral soluio of h diffrial quaio is p os si W ow appl h iiial odiios o fid ad. - 5 os π ) osπ siπ π 5π osπ ( Si siπ,osπ Thrfor 9π Now / 9π si os 5si 5os / Thrfor ( π ) 9π siπ osπ 5π siπ 5osπ 7. H h soluio of h iiial valu problm is 9π os 7si 5 os. // / Eampl 9 Solv h diffrial quaio Soluio: Th assoiad homogous diffrial quaio is // 6 / 9.Pu m.th h auiliar quaio is m 6m 9 m, Thus h omplmar fuio is Coprigh Virual Uivrsi of Pakisa

153 Diffrial Equaios (MTH) Si g( ) ( ) g( ) g ( ) W assum ha Corrspodig o g ( ) : p A B C Corrspodig o g ( ) : p D Thus h assumd form of h pariular soluio is Th fuio i p A B C D p is dupliad bw ad p. Mulipliaio wih dos o rmov his dupliaio. Howvr, if w mulipl rmovd. Thus h opraiv from of a pariular soluio is p A B C D p wih, his dupliaio is Th p A B D D ad p 6 A D D 9D Subsiuig io h giv diffrial quaio ad ollig lik rm, w obai // / p 6 p p 9A ( A 9B) A 6B 9C D 6 Equaig osa rms ad offiis of, ad A 6B 9C, A 9B 9A 6, D ilds Solvig hs quaios, w hav h valus of h ukow offiis Thus A, B 8 9, C ad D -6 p H h gral soluio p 6. 9 Highr Ordr Equaio Th mhod of udrmid offiis a also b usd for highr ordr quaios of h form d d d a a... a a g( ) d d d Coprigh Virual Uivrsi of Pakisa

154 Diffrial Equaios (MTH) wih osa offiis. Th ol rquirm is ha g() osiss of h propr kids of fuios as disussd arlir. Eampl Solv Soluio: /// // os To fid h omplmar fuio w solv h assoiad homogous diffrial /// // quaio Pu m m m, m, m Th h auiliar quaio is m m m ( m ) m,, Th auiliar quaio has qual ad disi ral roos. Thrfor, h omplmar fuio is Si Thrfor, w assum ha g( ) os p A os B si Clarl, hr is o dupliaio of rms bw ad p. Subsiuig h drivaivs of rms, w hav p i h giv diffrial quaio ad groupig h lik Equaig h offiis, of /// // p p ( A B) os ( A B) si Solvig hs quaios, w obai So ha a pariular soluio is os ad si, ilds A B, A B A /, B / 5 p (/) os (/ 5) si H h gral soluio of h giv diffrial quaio is p os. (/) os (/ 5) si Eampl Drmi h form of a pariular soluio of h DE Soluio Cosidr h assoiad homogous diffrial quaio Coprigh Virual Uivrsi of Pakisa

155 Diffrial Equaios (MTH) Th auiliar quaio is m m m,,, Thrfor, h omplmar fuio is Si g( ) g( ) g ( ) Corrspodig o g ( ) : p A Corrspodig o g ( ) : p B Thrfor, h ormal assumpio for h pariular soluio is Clarl hr is dupliaio of (i) (ii) p A B Th osa fuio bw ad p. Th poial fuio bw ad p. To rmov his dupliaio, w mulipl p wih ad p wih. This dupliaio a b rmovd b muliplig wih ad. H, h orr assumpio for h pariular soluio p is p A B 7.6 Eris Solv h followig diffrial quaios usig h udrmid offiis. // /. // /. 8 6 //. 8 // /. os // 5. ( )si // / // / 7. (os si ) Solv h followig iiial valu problms. // / / 8. ( ), ( ), ( ) 5 d / 9. ω F osγ, (), () d /// / // , ( ) 5, ( ), () Coprigh Virual Uivrsi of Pakisa

156 Diffrial Equaios (MTH) Rall 8 Udrmid Coffii (Aihilaor Opraor Approah). Tha a o-homogous liar diffrial quaio of ordr is a quaio of h form a d a d d d a a g( ) d d Th followig diffrial quaio is alld h assoiad homogous quaio d d d a a a a d d d Th offiis a, a,, a a b fuios of. Howvr, w will disuss quaios wih osa offiis.. Tha o obai h gral soluio of a o-homogous liar diffrial quaio w mus fid: Th omplmar fuio, whih is gral soluio of h assoiad homogous diffrial quaio. A pariular soluio of h o-homogous diffrial quaio. p. Tha h gral soluio of h o-homogous liar diffrial quaio is giv b Gral Soluio Complmar Fuio Pariular Igral Fidig h omplmar fuio has b ompll disussd i a arlir lur I h prvious lur, w sudid a mhod for fidig pariular igral of h o-homogous quaios. This was h mhod of udrmid offiis dvlopd from h viwpoi of suprposiio priipl. I h prs lur, w will lar o fid pariular igral of h ohomogous quaios b h sam mhod uilizig h op of diffrial aihilaor opraors. 8. Diffrial Opraors I alulus, h diffrial offii d / d is of dod b h apial lr D. So ha d D d Th smbol D is kow as diffrial opraor. This opraor rasforms a diffriabl fuio io aohr fuio,.g. Coprigh Virual Uivrsi of Pakisa 5

157 Diffrial Equaios (MTH) D( ), D(5 6 ) 5, D(os ) si Th diffrial opraor D posssss h propr of liari. This mas ha if f, g ar wo diffriabl fuios, h D { af ( ) bg( )} adf ( ) bdg( ) Whr a ad b ar osas. Baus of his propr, w sa ha D is a liar diffrial opraor. Highr ordr drivaivs a b prssd i rms of h opraor D i a aural mar: Similarl d d d d D( D) D d d d D d d,, d D Th followig polomial prssio of dgr ivolvig h opraor D a D ad ad a is also a liar diffrial opraor. For ampl, h followig prssios ar all liar diffrial opraors D, D D, 5D 6D D 8. Diffrial Equaio i Trms of D A liar diffrial quaio a b prssd i rms of h oaio D. Cosidr a d ordr quaio wih osa offiis Si // / a b g( ) d d D, D d d Thrfor h quaio a b wri as ad bd g( ) or ( ad bd ) g( ) Now, w dfi aohr diffrial opraor L as L ad bd Th h quaio a b ompal wri as L ( ) g( ) Coprigh Virual Uivrsi of Pakisa 6

158 Diffrial Equaios (MTH) Th opraor L is a sod-ordr liar diffrial opraor wih osa offiis. // / Eampl Cosidr h diffrial quaio 5 Si d d D, D d d Thrfor, h quaio a b wri as ( D D ) 5 Now, w dfi h opraor L as L D D Th h giv diffrial a b ompal wri as L ( ) 5 Faorizaio of a diffrial opraor A h-ordr liar diffrial opraor L a D ad ad a wih osa offiis a b faorizd, whvr h hararisis polomial quaio L a m am am a a b faorizd. Th faors of a liar diffrial opraor wih osa offiis ommu. Eampl (a) Cosidr h followig d ordr liar diffrial opraor (b) D 5D 6 If w ra D as a algbrai quai, h h opraor a b faorizd as D 5D 6 ( D )( D ) To illusra h ommuaiv propr of h faors, w osidr a widiffriabl fuio f (). Th w a wri To vrif his w l ( D 5D 6) ( D )( D ) ( D )( D ) w ( D ) Th ( D ) w Dw w ( D ) w ( ) ( 6) // / / // / // / ( D ) w 5 6 ( D )( D ) 5 6 Coprigh Virual Uivrsi of Pakisa 7

159 Diffrial Equaios (MTH) Similarl if w l / w ( D ) ( ) // / / Th ( D ) w Dw w ( ) ( 6) or // / ( D ) w 5 6 or // / ( D )( D ) 5 6 Thrfor, w a wri from h wo prssios ha ( D )( D ) ( D )( D ) H Eampl ( D )( D ) ( D )( D ) (a) Th opraor D a b faorizd as D ( D )( D ). or D ( D -)( D ) (b) Th opraor D D dos o faor wih ral umbrs. Eampl Th diffrial quaio D ( D )( D ) ( D ). a b wri as ( D ) 8. Aihilaor Opraor Suppos ha L is a liar diffrial opraor wih osa offiis. f() dfis a suffiil diffriabl fuio. Th fuio f is suh ha L() Th h diffrial opraor L is said o b a aihilaor opraor of h fuio f. Eampl 5 Si D, D, D, D, Thrfor, h diffrial opraors D, D, D, D, ar aihilaor opraors of h followig fuio k (a osa),,,, I gral, h diffrial opraor,,,, D aihilas ah of h fuios Coprigh Virual Uivrsi of Pakisa 8

160 Diffrial Equaios (MTH) H, w olud ha h polomial fuio a b aihilad b fidig a opraor ha aihilas h highs powr of. Eampl 6 Fid a diffrial opraor ha aihilas h polomial fuio 5 8. Soluio Si, H, D D D ( 5 8 ). D is h diffrial opraor ha aihilas h fuio. No ha h fuios ha ar aihilad b a h-ordr liar diffrial opraor L ar simpl hos fuios ha a b obaid from h gral soluio of h homogous diffrial quaio L ( ). Eampl 7 Cosidr h homogous liar diffrial quaio of ordr ( D α ).Th auiliar quaio of h diffrial quaio is ( m α ) m α, α,, α ( ims) Thrfor, h auiliar quaio has a ral roo α of muliplii. So ha h diffrial quaio has h followig liarl idpd soluios: α α,, α α,, Thrfor, h gral soluio of h diffrial quaio is So ha h diffrial opraor ( D α) aihilas ah of h fuios α α α α α, α, α,,. H, as a osqu of h fa ha h diffriaio a b prformd rm b rm, h diffrial opraor ( D α) α α α α α aihilas h fuio Eampl 8 5 Fid a aihilaor opraor for h fuios:(a) f ( ), (b) g( ) 6 Soluio (a) Si ( D 5) 5 5. Thrfor, h aihilaor opraor of fuio f is giv b L D 5 Coprigh Virual Uivrsi of Pakisa 9

161 Diffrial Equaios (MTH) W oi ha i his asα 5,. (b) Similarl ( D ) ( 6 ) ( D D )( ) ( D D )(6 ) D or ( ) ( ) or ( D ) ( 6 ) Thrfor, h aihilaor opraor of h fuio g is giv b L ( D ) W oi ha i his as α. Eampl 9 Cosidr h diffrial quaio ( D α D ( α β ) Th auiliar quaio is ( m αm ( α β ) m αm ( α β ) Thrfor, wh α, β ar ral umbrs, w hav from h quadrai formula ( α β ) α β α ± α m ± i Thrfor, h auiliar quaio has h followig wo ompl roos of muliplii. m α iβ, m α iβ Thus, h gral soluio of h diffrial quaio is a liar ombiaio of h followig liarl idpd soluios α os β, α os β, α os β,, α os β α si β, α si β, α si β,, α si β H, h diffrial opraor ( D α D ( α β ) is h aihilaor opraor of h fuios α os β, α os β, α os β,, α os β α si β, α si β, α si β,, α si β Eampl If w ak α, β, Th h diffrial opraor ( D α D ( α β ) Also, i a b vrifid ha ( D D 5) os boms D D 5.. Coprigh Virual Uivrsi of Pakisa 5

162 Diffrial Equaios (MTH) Thrfor, h liar diffrial opraor D D 5 ( ) ( ) os aihilas h fuios si Now, osidr h diffrial quaio Th auiliar quaio is Thrfor, h fuios ( D D 5) m m 5 m ± i ( ) os ( ) si ar h wo liarl idpd soluios of h diffrial quaio ( ) D D 5, Thrfor, h opraor also aihilas a liar ombiaio of ad,.g os 9 si. Eampl If w ak α, β, Th h diffrial opraor ( D α D ( α β ) Boms ( D ) D D Also, i a b vrifid ha ad ( D D ) os ( D D ) si ( D D ) os ( D D ) si Thrfor, h liar diffrial opraor D D aihilas h fuios os, si os, si Coprigh Virual Uivrsi of Pakisa 5

163 Diffrial Equaios (MTH) Eampl Takig, D β α, h opraor ( D α D ( α β ) Si ( D β ) os β β os β β os β ( D β ) si β β si β β si β Thrfor, h diffrial opraor aihilas h fuios No ha f ( ) os β, g( ) si β boms If a liar diffrial opraor wih osa offiis is suh ha L ( ), L ( ) i.. h opraor L aihilas h fuios ad. Th h opraor L aihilas hir liar ombiaio. [ ( ) ( ) ] L. This rsul follows from h liari propr of h diffrial opraor L. Suppos ha L ad L ar liar opraors wih osa offiis suh ha L ( ), L ( ) ad ( ), ( ) L L h h produ of hs diffrial opraors L L aihilas h liar sum ( ) ( ) So ha L L [ ( ) ( ) ] To dmosra his fa w us h liari propr for wriig ( ) L L ( ) L L ( ) L L Si L L LL hrfor L L ( ) L L ( ) L L ( ) or L L ( ) L L ( )] L [ L ( )] [ Bu w kow ha L ( ), L ( ) Thrfor L L ( ) L ] L [] [ Coprigh Virual Uivrsi of Pakisa 5

164 Diffrial Equaios (MTH) Eampl Fid a diffrial opraor ha aihilas h fuio f ( ) 7 6si Soluio Suppos ha ) 7, ( ) 6si ( ( ) D ( ) D 7 ( D 9) ( ) ( D 9) si Thrfor, D ( D 9) aihilas h fuio f (). Eampl Fid a diffrial opraor ha aihilas h fuio f ( ) Soluio Suppos ha ( ) ( ) D D, ( D ) ( D ). ( ), ( ) Thrfor, h produ of wo opraors ( )( ) aihilas h giv fuio No ha D D f ( ) Th diffrial opraor ha aihilas a fuio is o uiqu. For ampl, ( 5 D 5), ( 5)( 5 D D ), 5 ( D 5) D Thrfor, hr ar aihilaor opraors of h fuios, aml D 5 D, ( D 5) D ( D 5), ( )( ) Wh w sk a diffrial aihilaor for a fuio, w wa h opraor of lows possibl ordr ha dos h job. 8. Eris Wri h giv diffrial quaio i h form ( ) g( ), opraor wih osa offiis.. d 5 9si d. d 8 d L whr L is a diffrial Coprigh Virual Uivrsi of Pakisa 5

165 Diffrial Equaios (MTH). d d d 5 d d d. d d d 7 6 si d d d Faor h giv diffriabl opraor, if possibl. 5. 9D D D D D 8D 6. D Vrif ha h giv diffrial opraor aihilas h idiad fuios / 9. D ;. D 6; os8-5si 8 Fid a diffrial opraor ha aihilas h giv fuio... si 6 Coprigh Virual Uivrsi of Pakisa 5

166 Diffrial Equaios (MTH) 9 Udrmid Coffiis(Aihilaor Opraor Approah) Th mhod of udrmid offiis ha uilizs h op of aihilaor opraor approah is also limid o o-homogous liar diffrial quaios Tha hav osa offiis, ad Whr h fuio g () has a spifi form. Th form of g () :Th ipu fuio g () has o hav o of h followig forms: A osa fuio k. A polomial fuio A poial fuio Th rigoomri fuios si( β ), os( β ) Fii sums ad produs of hs fuios. Ohrwis, w ao appl h mhod of udrmid offiis. 9. Soluio Mhod Cosidr h followig o-homogous liar diffrial quaio wih osa offiis of ordr d d d a a a a g( ) d d d If L dos h followig diffrial opraor L a D ad ad a Th h o-homogous liar diffrial quaio of ordr a b wri as L ( ) g( ) Th fuio g() should osis of fii sums ad produs of h propr kid of fuios as alrad plaid. Th mhod of udrmid offiis, aihilaor opraor approah, for fidig a pariular igral of h o-homogous quaio osiss of h followig sps: Sp Wri h giv o-homogous liar diffrial quaio i h form L ( ) g( ) Sp Fid h omplmar soluio b fidig h gral soluio of h assoiad homogous diffrial quaio: L ( ) Sp Opra o boh sids of h o-homogous quaio wih a diffrial opraor L ha aihilas h fuio g(). Coprigh Virual Uivrsi of Pakisa 55

167 Diffrial Equaios (MTH) Sp Fid h gral soluio of h highr-ordr homogous diffrial quaio L L( ) Sp 5 Dl all hos rms from h soluio i sp ha ar dupliad i h omplmar soluio, foud i sp. Sp 6 Form a liar ombiaio p of h rms ha rmai. This is h form of a pariular soluio of h o-homogous diffrial quaio Sp 7 Subsiu quaio L () g( ) p foud i sp 6 io h giv o-homogous liar diffrial L ( ) g( ) Mah offiis of various fuios o ah sid of h quali ad solv h rsulig ssm of quaios for h ukow offiis i p. Sp 8 Wih h pariular igral foud i sp 7, form h gral soluio of h giv diffrial quaio as: Eampl Solv Soluio: d d d d. Sp Si d d D, D d d Thrfor, h giv diffrial quaio a b wri as ( D D ) Sp To fid h omplmar fuio, w osidr h assoiad homogous diffrial quaio Th auiliar quaio is ( D D ) p m m ( m )( m ) m, Thrfor, h auiliar quaio has wo disi ral roos. m, m, Thus, h omplmar fuio is giv b Coprigh Virual Uivrsi of Pakisa 56

168 Diffrial Equaios (MTH) Sp I his as h ipu fuio is g ( ) Furhr D g( ) D Thrfor, h diffrial opraor D aihilas h fuio g. Opraig o boh sids of h quaio i sp, w hav D ( D D ) D D ( D D ) This is h homogous quaio of ordr 5. N w solv his highr ordr quaio. Sp Th auiliar quaio of h diffrial quaio i sp is m ( m m ) m ( m )( m ) m,,,, Thus is gral soluio of h diffrial quaio mus b Sp 5 Th followig rms osiu 5 5 Thrfor, w rmov hs rms ad h rmaiig rms ar Sp 6 This mas ha h basi sruur of h pariular soluio p is p A B C, Whr h osas, ad hav b rplad, wih A, B, ad C, rspivl. Sp 7 Si p A B C p B C, p C Thrfor p p p C B 6C A B C or p p p (C) (B 6C) (A B C) Coprigh Virual Uivrsi of Pakisa 57

169 Diffrial Equaios (MTH) Subsiuig io h giv diffrial quaio, w hav Equaig h offiis of (C) (B 6C) (A B C), C B Solvig hs quaios, w obai ad h osa rms, w hav 6C A B C A 7, B 6, C H p 7 6 Sp 8 Th gral soluio of h giv o-homogous diffrial quaio is p 7 6. Eampl Solv Soluio: Sp Si d d 8 d d si d d D, D d d Thrfor, h giv diffrial quaio a b wri as ( D D) 8 si Sp W firs osidr h assoiad homogous diffrial quaio o fid Th auiliar quaio is m ( m ) m, Thus h auiliar quaio has ral ad disi roos. So ha w hav Sp I his as h ipu fuio is giv b g( ) 8 si Si ( D )(8 ), ( D )(si ) Coprigh Virual Uivrsi of Pakisa 58

170 Diffrial Equaios (MTH) Thrfor, h opraors D ad D aihila 8 ad si, rspivl. So h opraor ( D )( D ) aihilas h ipu fuio g (). This mas ha ( D )( D ) g( ) ( D )( D )(8 si ) W appl ( D )( D ) o boh sids of h diffrial quaio i sp o obai ( D )( D )( D D). This is homogous diffrial quaio of ordr 5. Sp Th auiliar quaio of h highr ordr quaio foud i sp is ( m )( m m ( m ) ( m )( m ) m,,, ± i m) Thus, h gral soluio of h diffrial quaio os 5 si Sp 5 Firs wo rms i his soluio ar alrad prs i Thrfor, w limia hs rms. Th rmaiig rms ar os 5 si Sp 6 Thrfor, h basi sruur of h pariular soluio p A B os C si p mus b Th osas, 5 hav b rplad wih h osas rspivl. Sp 7 Si p A B os C si A, B ad C, Thrfor p p A ( B C) os ( B C)si Subsiuig io h giv diffrial quaio, w hav A ( B C) os ( B C)si 8 si. Equaig offiis of, os Solvig hs quaios w obai ad si, w obai A 8, B C, B C Coprigh Virual Uivrsi of Pakisa 59

171 Diffrial Equaios (MTH) A 8/, B 6/5, C /5 p 8 6 os si. 5 5 Sp 8 Th gral soluio of h diffrial quaio is h Eampl Solv Soluio d 8 5. d 8 6 os si. 5 5 Sp Th giv diffrial quaio a b wri as ( D 8) 5 Sp Th assoiad homogous diffrial quaio is ( D 8) Roos of h auiliar quaio ar ompl m ± i Thrfor, h omplmar fuio is os si Sp Si D, ( D ) Thrfor h opraors D ad D aihila h fuios 5 ad. W appl D ( D ) o h o-homogous diffrial quaio D ( D )( D 8). This is a homogous diffrial quaio of ordr 5. Sp Th auiliar quaio of his diffrial quaio is m ( m )( m 8) m,,, ± i Thrfor, h gral soluio of his quaio mus b os si 5 Coprigh Virual Uivrsi of Pakisa 6

172 Diffrial Equaios (MTH) Sp 5 Si h followig rms ar alrad prs i os si Thus w rmov hs rms. Th rmaiig os ar 5 Sp 6 Th basi form of h pariular soluio of h quaio is p A B C Th osas, ad 5 hav b rplad wih A, B ad C. Sp 7 Si p A B C Thrfor p 8 p 8A 8B 9C Subsiuig i h giv diffrial quaio, w hav Equaig offiis of Thus 8A 8B 9C 5, ad h osa rms, w hav A, B 5/ 8, C p / 9 Sp 8 H, h gral soluio of h giv diffrial quaio is p 5 or os si. 8 9 d Eampl Solv os os d Soluio: Sp Th giv diffrial quaio a b wri as ( D ) os os Sp Cosidr h assoiad diffrial quaio Coprigh Virual Uivrsi of Pakisa 6

173 Diffrial Equaios (MTH) Th auiliar quaio is Thrfor ( D ) m m ± i os si Sp Si ( D ) ( os ) Thrfor, h opraor ( D ) os ; ( D ) os os aihilas h ipu fuio Thus opraig o boh sids of h o-homogous quaio wih ( D ), w hav ( D ) ( D ) or ( D ) This is a homogous quaio of ordr 6. Sp Th auiliar quaio of his highr ordr diffrial quaio is ( m ) m i, i, i, i, i, i Thrfor, h auiliar quaio has ompl roos i, ad olud ha os si os si os Sp 5 Si firs wo rms i h abov soluio ar alrad prs i os si Thrfor, w rmov hs rms. Sp 6 Th basi form of h pariular soluio is p A os Bsi C os E si i boh of muliplii. W 5 6 si Sp 7 Si p A os Bsi C os E si Thrfor p p E os Csi (B C)os ( A E) si Subsiuig i h giv diffrial quaio, w obai E os Csi (B C)os ( A E)si os os Equaig offiis of os, si, os ad si, w obai Coprigh Virual Uivrsi of Pakisa 6

174 Diffrial Equaios (MTH) E, C B C, A E Solvig hs quaios w obai A /, B /, C, E / Thus p os si si Sp 8 H h gral soluio of h diffrial quaio is os si os si si. Eampl 5 Drmi h form of a pariular soluio for d d d d os Soluio Sp Th giv diffrial quaio a b wri as ( D D ) os Sp To fid h omplmar fuio, w osidr Th auiliar quaio is m m ( m ) m, Th omplmar fuio for h giv quaio is Sp Si ( D D 5) os Applig h opraor ( D D 5) o boh sids of h quaio, w hav ( D D 5)( D D ) This is homogous diffrial quaio of ordr. Sp Th auiliar quaio is ( m m 5)( m m ) m ± i,, Thrfor, gral soluio of h h ordr homogous quaio is os si Coprigh Virual Uivrsi of Pakisa 6

175 Diffrial Equaios (MTH) Sp 5 Si h rms ar alrad prs i, hrfor, w rmov hs ad h rmaiig rms ar os si Sp 6 Thrfor, h form of h pariular soluio of h o-homogous quaio is p A os B si No ha h sps 7 ad 8 ar o dd, as w do hav o solv h giv diffrial quaio. Eampl 6 Drmi h form of a pariular soluio for Soluio: d d d d Sp Th giv diffrial a b rwri as. d d 5 ( D D D) 5 6 Sp To fid h omplmar fuio, w osidr h quaio Th auiliar quaio is ( D D D) m m m m ( m m ) m ( m ) m Thus h omplmar fuio is,, Sp Si 5 g( ) 5 6 Furhr D (5 6) ( D ) ( D 5) 5 Thrfor h followig opraor mus aihila h ipu fuio g (). Thrfor, applig h opraor D ( D ) ( D 5) o boh sids of h o-homogous quaio, w hav D ( D ) ( D 5)( D D 5 or D ( D ) ( D 5) D) Coprigh Virual Uivrsi of Pakisa 6

176 Diffrial Equaios (MTH) This is homogous diffrial quaio of ordr. Sp Th auiliar quaio for h h ordr diffrial quaio is 5 m ( m ) ( m 5) m,,,,,,,,, 5 H h gral soluio of h h ordr quaio is Sp 5 Si h followig rms osiu h omplmar fuio, w rmov hs Thus h rmaiig rms ar H, h form of h pariular soluio of h giv quaio is 5 p A B C E F G H \ 9. Eris Solv h giv diffrial quaio b h udrmid offiis os si os si os si, ( π / ), ( π / ) 9. 5, (), ( ), ( ) ()., (), ( ), ( ), ( ) Coprigh Virual Uivrsi of Pakisa 65

177 Diffrial Equaios (MTH) Rall No ha Variaio of Paramrs Tha a o-homogous liar diffrial quaio wih osa offiis is a quaio of h form a d a d d d a a g( ) d d Th gral soluio of suh a quaio is giv b Gral Soluio Complmar Fuio Pariular Igral Fidig h omplmar fuio has alrad b ompll disussd. I h las wo lurs, w lar how o fid h pariular igral of h ohomogous quaios b usig h udrmid offiis. Tha h gral soluio of a liar firs ordr diffrial quaio of h form d d P ( ) f ( ) Pd Pd Pd. f d is giv b ( ) I his las quaio, h d rm Pd is soluio of h assoiad homogous quaio: Similarl, h s rm d d. ( ) P Pd Pd p. f ( )d is a pariular soluio of h firs ordr o-homogous liar quaio. diffrial Thrfor, h soluio of h firs ordr liar diffrial quaio a b wri i h form p I his lur, w will us h variaio of paramrs o fid h pariular igral of h o-homogous quaio. Th Variaio of Paramrs Coprigh Virual Uivrsi of Pakisa 66

178 Diffrial Equaios (MTH). Firs ordr quaio Th pariular soluio p of h firs ordr liar diffrial quaio is giv b. Pd Pd p. f ( )d This formula a also b drivd b aohr mhod, kow as h variaio of paramrs. Th basi produr is sam as disussd i h lur o osruio of a sod soluio Si Pd is h soluio of h homogous diffrial quaio d P( ), d ad h quaio is liar. Thrfor, h gral soluio of h quaio is Th variaio of paramrs osiss of fidig a fuio u ( ) suh ha u ( ) ( ) p is a pariular soluio of h o-homogous diffrial quaio ( ) d P d ( ) f ( ) Noi ha h paramr has b rplad b h variabl u. W subsiu h giv quaio o obai d du u P d d ( ) f ( ) Si is a soluio of h o-homogous diffrial quaio. Thrfor w mus hav d P( ) d So ha w obai du f ( ) d This is a variabl sparabl quaio. B sparaig h variabls, w hav f ( ) du d ( ) p i Coprigh Virual Uivrsi of Pakisa 67

179 Diffrial Equaios (MTH) Igraig h las prssio w.r.o, w obai Thrfor, h pariular soluio ( ) f Pd u( ) d f ( ) d p of h giv firs-ordr diffrial quaio is. u( ) Pd Pd d or.. f ( ) p u. Sod Ordr Equaio f ( ) ( ) Cosidr h d ordr liar o-homogous diffrial quaio d ( ) a ( ) a ( ) g( ) a B dividig wih a ( ), w a wri his quaio i h sadard form Th fuios P( ), Q( ) f ( ) ( ) Q( ) f ( ) P ad ar oiuous o som irval I. For h omplmar fuio w osidr h assoiad homogous diffrial quaio P Q Complmar fuio ( ) ( ) Suppos ha ad ar wo liarl idpd soluios of h homogous quaio. Th ad form a fudamal s of soluios of h homogous quaio o h irval I. Thus h omplmar fuio is ( ) ( ) Si ad ar soluios of h homogous quaio. Thrfor, w hav P Q Pariular Igral ( ) ( ) ( ) Q ( ) P For fidig a pariular soluio, w rpla h paramrs p ad i h omplmar fuio wih h ukow variabls u ( ) ad u ( ). So ha h assumd pariular igral is Coprigh Virual Uivrsi of Pakisa 68

180 Diffrial Equaios (MTH) p ( ) ( ) ( ) ( ) u u Si w sk o drmi wo ukow fuios u ad u, w d wo quaios ivolvig hs ukows. O of hs wo quaios rsuls from subsiuig h assumd p i h giv diffrial quaio. W impos h ohr quaio o simplif h firs drivaiv ad hrb h d drivaiv of p. p u u u u u u u u To avoid d drivaivs of u adu, w impos h odiio u u Th p u u So ha Thrfor p u u p P u u p Q p u Pu u Subsiuig i h giv o-homogous diffrial quaio ilds Pu u Qu ` Qu u u u u u Pu Pu Qu Qu f ( ) ` or u P Q ] u [ P Q ] u u f ( ) [ Now makig us of h rlaios w obai ( ) Q ( ) ( ) Q ( ) P P u u H u ad u mus b fuios ha saisf h quaios u u f ( ) u f u B usig h Cramr s rul, h soluio of his s of quaios is giv b ( ) Coprigh Virual Uivrsi of Pakisa 69

181 Diffrial Equaios (MTH) W u, W u W W WhrW, W ad W do h followig drmias W, W, W f f ( ) ( ) Th drmia W a b idifid as h Wroskia of h soluios ad. Si h soluios ad ar liarl idpd o I. Thrfor W ( ( ) ( ) ),., I Now igraig h prssios for u ad u, w obai h valus of u ad u, h h pariular soluio of h o-homogous liar diffrial quaio.. Summar of h Mhod To solv h d ordr o-homogous liar diffrial quaio a a a g( ), usig h variaio of paramrs, w d o prform h followig sps: Sp W fid h omplmar fuio b solvig h assoiad homogous diffrial quaio a a a Sp If h omplmar fuio of h quaio is giv b h ad ar wo liarl idpd soluios of h homogous diffrial quaio. Th ompu h Wroskia of hs soluios. W Sp B dividig wih a, w rasform h giv o-homogous quaio io h sadard form P Q f ad w idif h fuio f ( ). ( ) ( ) ( ) Sp W ow osru h drmias W adw giv b W, f ( ) W f ( ) Coprigh Virual Uivrsi of Pakisa 7

182 Diffrial Equaios (MTH) Sp 5 N w drmi h drivaivs of h ukow variabls u ad u hrough h rlaios W W u, u W W Sp 6 Igra h drivaivs u ad u o fid h ukow variabls u ad u. So ha W W u d, u d W W Sp 7 Wri a pariular soluio of h giv o-homogous quaio as p u u Sp 8 Th gral soluio of h diffrial quaio is h giv b.. Cosas of Igraio p u u. W do d o irodu h osas of igraio, wh ompuig h idfii igrals i sp 6 o fid h ukow fuios of u ad u. For, if w do irodu hs osas, h ( u a ) ( u b) p So ha h gral soluio of h giv o-homogous diffrial quaio is or ( ) ( ) ( u a ) ( u b ) p a b u u If w rpla a wih C ad bwihc, w obai C C u u This dos o provid ahig w ad is similar o h gral soluio foud i sp 8, aml Eampl u u. Solv ( ) Soluio: Sp To fid h omplmar fuio Coprigh Virual Uivrsi of Pakisa 7

183 Diffrial Equaios (MTH) Pu Th h auiliar quaio is m m, m, m m m m ( ) Rpad ral roos of h auiliar quaio m m, Sp B h ispio of h omplmar fuio, w mak h idifiaio W W ad Thrfor (, ) (, ), Sp Th giv diffrial quaio is ( ) Si his quaio is alrad i h sadard form P Q Thrfor, w idif h fuio f () as f ( ) ( ) f ( ) ( ) ( ) Sp W ow osru h drmias W ( ) ( ) W ( ) ( ) Sp 5 W drmi h drivaivs of h fuios u ad u i his sp ( ) W u W W ( ) u W Sp 6 Igraig h las wo prssios, w obai Coprigh Virual Uivrsi of Pakisa 7

184 Diffrial Equaios (MTH) u u ( ( ) d ) d. Rmmbr! W do hav o add h osas of igraio. Sp 7 Thrfor, a pariular soluio of h giv diffrial quaio is or p p 6 Sp 8 H, h gral soluio of h giv diffrial quaio is p 6 Eampl Solv 6 s. Soluio: Sp To fid h omplmar fuio w solv h assoiad homogous diffrial quaio 6 9 Th auiliar quaio is m 9 m ± i Roos of h auiliar quaio ar ompl. Thrfor, h omplmar fuio is os si Sp From h omplmar fuio, w idif os, si as wo liarl idpd soluios of h assoiad homogous quaio. Thrfor os si W ( os,si ) si os Sp B dividig wih, w pu h giv quaio i h followig sadard form Coprigh Virual Uivrsi of Pakisa 7

185 Diffrial Equaios (MTH) So ha w idif h fuio f () as 9 s. f ( ) s Sp W ow osru h drmias W ad W si W s si s os W os os si s si Sp 5 Thrfor, h drivaivs u ad u ar giv b u W W, W W u os si Sp 6 Igraig h las wo quaios w.r.o, w obai u ad u l si 6 No ha o osas of igraio hav b addd. Sp 7 Th pariular soluio of h o-homogous quaio is p os ( si ) l si 6 Sp 8 H, h gral soluio of h giv diffrial quaio is p os si os ( si ) l si 6 Eampl Solv. Soluio: Sp For h omplmar fuio osidr h assoiad homogous quaio Coprigh Virual Uivrsi of Pakisa 7

186 Diffrial Equaios (MTH) To solv his quaio w pu Th h auiliar quaio is: m m m, m, m m m ± Th roos of h auiliar quaio ar ral ad disi. Thrfor, h omplmar fuio is Sp From h omplmar fuio w fid, Th fuios ad ar wo liarl idpd soluios of h homogous quaio. Th Wroskia of hs soluios is W (, ) Sp Th giv quaio is alrad i h sadard form Hr f ( ) ( ) ( ) ( ) p Q f Sp W ow form h drmias W / W / (/ ) (/ ) Sp 5 Thrfor, h drivaivs of h ukow fuios uad u ar giv b W u W ( / ) Coprigh Virual Uivrsi of Pakisa 75

187 Diffrial Equaios (MTH) ( ) W W u / Sp 6 W igra hs wo quaios o fid h ukow fuios u ad u. u d, u d Th igrals dfiig ad u u ao b prssd i rms of h lmar fuios ad i is usomar o wri suh igral as:, - u d u d Sp 7 A pariular soluio of h o-homogous quaios is p d d Sp 8 H, h gral soluio of h giv diffrial quaio is d d p Coprigh Virual Uivrsi of Pakisa 76

188 Diffrial Equaios (MTH) Variaio of Paramrs Mhod for Highr-Ordr Equaios Th mhod of h variaio of paramrs jus amid for sod-ordr diffrial quaios a b gralizd for a h-ordr quaio of h p. a d a d d d a a g( ) d d Th appliaio of h mhod o h ordr diffrial quaios osiss of prformig h followig sps. Sp To fid h omplmar fuio w solv h assoiad homogous quaio d d d a a a a d d d Sp Suppos ha h omplmar fuio for h quaio is Th,,, ar liarl idpd soluios of h homogous quaio. Thrfor, w ompu Wroskia of hs soluios. W(,,,, ) Sp W wri h diffrial quaio i h form ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) P P P f ad ompu h drmias W k ; k,,, ; b rplaig h kh olum of W b h olum f ( ) Sp 5 N w fid h drivaivs hrough h rlaios u, u,, u of h ukow fuios u, u,, u Coprigh Virual Uivrsi of Pakisa 77

189 Diffrial Equaios (MTH) u k Wk W, k,,, No ha hs drivaivs a b foud b solvig h quaios u u u u u u ( ) u ( ) u ( ) u f ( ) Sp 6 Igra h drivaiv fuios ompud i h sp 5 o fid h fuios u k Wk uk d, k,,, W Sp 7 W wri a pariular soluio of h giv o-homogous quaio as u u u ( ) ( ) ( ) ( ) ( ) ( ) p Sp 8 Havig foud h omplmar fuio ad h pariular igral p, w wri h gral soluio b subsiuio i h prssio: p No ha Th firs quaios i sp 5 ar assumpios mad o simplif h firs drivaivs of p. Th las quaio i h ssm rsuls from subsiuig h pariular igral p ad is drivaivs io h giv h ordr liar diffrial quaio ad h simplifig. Dpdig upo how h igrals of h drivaivs u k of h ukow fuios ar foud, h aswr for p ma b diffr for diffr amps o fid p for h sam quaio. Wh askd o solv a iiial valu problm, w d o b sur o appl h iiial odiios o h gral soluio ad o o h omplmar fuio alo, hikig ha i is ol ha ivolvs h arbirar osas. Eampl Solv h diffrial quaio b variaio of paramrs. d d Soluio: Sp Th assoiad homogous quaio is d d Auiliar quaio m m m ( m ) m, m ± i Thrfor h omplmar fuio is os si d d s d d Coprigh Virual Uivrsi of Pakisa 78

190 Diffrial Equaios (MTH) Sp : Si os si, os, si So ha h Wroskia of h soluios, ad ( ) os si W,, si os os si B h lmar row opraio R R, w hav si os ( ) os si si os Sp : Th giv diffrial quaio is alrad i h rquird sadard form s Sp : N w fid h drmias W, W adw b rspivl, rplaig s, d ad rd olum of W b h olum W W s s os si os si os si ( ) s si os s si os s si os os s o s si ad W os si os s si si s os s Coprigh Virual Uivrsi of Pakisa 79

191 Diffrial Equaios (MTH) Sp 5: W ompu h drivaivs of h fuios u, u ad u as: W u W u W W s u W W o Sp 6: Igra hs drivaivs o fid u, u ad u u W d s d l s o W W os u d o d d l si W si W u d d W Sp 7: A pariular soluio of h o-homogous quaio is l s o os l si si p Sp 8: Th gral soluio of h giv diffrial quaio is: os si l so os l si si Eampl Solv h diffrial quaio b variaio of paramrs. a Soluio Sp : W fid h omplmar fuio b solvig h assoiad homogous quaio Corrspodig auiliar quaio is m ( m ) m m m, m ± i Thrfor h omplmar fuio is Sp : Si os si Coprigh Virual Uivrsi of Pakisa 8

192 Diffrial Equaios (MTH) Thrfor os si, os, Now w ompu h Wroskia of, ad ( ) B h lmar row opraio R R, w hav si os si W,, si os os si si os ( ) os si si os Sp : Th giv diffrial quaio is alrad i h rquird sadard form a Sp : Th drmias W, W adw ar foud b rplaig h s, d ad rd olum of W b h olum a Thrfor os si W si os W a os si ( ) a os si a si os a si ( os a ) si ad W os si ( si a ) si a os a Sp 5: W ompu h drivaivs of h fuios u, u ad u. Coprigh Virual Uivrsi of Pakisa 8

193 Diffrial Equaios (MTH) u W W a u W W si u W W si a Sp 6: W igra hs drivaivs o fid u, u ad u W W si os u d a d d l os W u d si d os W W u d si a d W si si d os si sd d d ( os ) s ( os s s ) ( ) os s d os d sd si l s a Sp 7: Thus, a pariular soluio of h o-homogous quaio ( ) ( ) l os os os si l s a si p l os os si si l s a l os si l s a Sp 8: H, h gral soluio of h giv diffrial quaio is: os si l os si l s a or ( ) os si l os si l s a or whr d rprss. d os si l os si l s a Eampl Solv h diffrial quaio b variaio of paramrs. Coprigh Virual Uivrsi of Pakisa 8

194 Diffrial Equaios (MTH) Soluio Sp : Th assoiad homogous quaio is Th auiliar quaio of h homogous diffrial quaio is m m m ( m ) ( m) m,, Th roos of h auiliar quaio ar ral ad disi. Thrfor is giv b Sp : From w fid ha hr liarl idpd soluios of h homogous diffrial quaio.,, Thus h Wroskia of h soluios, ad is giv b W B applig h row opraios RR, R R, w obai W 6 Sp : Th giv diffrial quaio is alrad i h rquird sadard form Sp : N w fid h drmias W, W adw b, rspivl, rplaig h s, d ad rd olum of W b h olum Coprigh Virual Uivrsi of Pakisa 8

195 Diffrial Equaios (MTH) W ( ) ( ) W ( ) ad ( ) W ( ) 6 Sp 5: Thrfor, h drivaivs of h ukow fuios u, u ad u ar giv b. W u W 6 W u W 6 u W W Sp 6: Igra hs drivaivs o fid u, u ad u W u d d d W W u d d W Coprigh Virual Uivrsi of Pakisa 8

196 Diffrial Equaios (MTH) u W d W d 6 Sp 7: A pariular soluio of h o-homogous quaio is p Sp 8: Th gral soluio of h giv diffrial quaio is:. Eris Solv h diffrial quaios b variaios of paramrs.. a. s a. s. 9 / 5. /( ) 6. / 7. s 8. 6 Solv h iiial valu problms ( ) Coprigh Virual Uivrsi of Pakisa 85

197 Diffrial Equaios (MTH) Appliaios of Sod Ordr Diffrial Equaio A sigl diffrial quaio a srv as mahmaial modl for ma diffr phoma i si ad girig. Diffr forms of h d ordr liar diffrial quaio d d a b f ( ) d d appar i h aalsis of problms i phsis, hmisr ad biolog. I h prs ad lur w shall fous o o appliaio; h moio of a mass aahd o a sprig. d W shall s, wha h idividual rms d a b, d d ad f ( ) mas i h o of vibraioal ssm. Ep for h rmiolog ad phsial irpraio of h rms d a, d b,, f ( ) d d h mahmais of a sris irui is idial o ha of a vibraig sprig-mass ssm. Thrfor w will disuss a LRC irui i lur.. Simpl Harmoi Moio Wh h Nwo s d law is ombid wih h Hook s Law, w a driv a diffrial quaio govrig h moio of a mass aahd o sprig h simpl harmoi moio... Hook s Law Suppos ha A mass is aahd o a flibl sprig suspdd from a rigid suppor, h Th sprig srhs b a amou s. Th sprig rs a rsorig F opposi o h dirio of logaio or srh. Th Hook s law sas ha h for F is proporioal o h logaio s. i. F ks Whr k is osa of proporioali, ad is alld sprig osa. No Tha Diffr masss srh a sprig b diffr amou i. s is diffr for diffr m. Th sprig is hararizd b h sprig osa k. For ampl if W lbs ad s f Th F ks or k Coprigh Virual Uivrsi of Pakisa 86

198 Diffrial Equaios (MTH) or If k lbs/f W 8 lbs h 8 ( s) s / 5 f.. Nwo s Sod Law Wh a for F as upo a bod, h alraio a is produd i h dirio of h for whos magiud is proporioal o h magiud of for. i. F ma Whr m is osa of proporioali ad i rprss mass of h bod... Wigh Th graviaioal for rd b h arh o a bod of mass m is alld wigh of h bod, dod b W I h abs of air rsisa, h ol for aig o a frl fallig bod is is wigh. Thus from Nwo s d law of moio W mg Whr m is masurd i slugs, kilograms or grams ad g f/s, 9.8m / s or 98 m/s... Diffrial Equaio Wh a bod of mass m is aahd o a sprig Th sprig srhs b a amou s ad aais a quilibrium posiio. A h quilibrium posiio, h wigh is balad b h rsorig for ks. Thus, h odiio of quilibrium is mg ks mg ks If h mass is displad b a amou from is quilibrium posiio ad h rlasd. Th rsorig for boms k(s ). So ha h rsula of wigh ad h rsorig for aig o h bod is giv b Rsula k ( s ) mg. B Nwo s d Law of moio, w a wri d m k( s ) mg d d or m k ks mg d Si mg ks d Thrfor m k d Th gaiv idias ha h rsorig for of h sprig as opposi o h dirio of moio. Coprigh Virual Uivrsi of Pakisa 87

199 Diffrial Equaios (MTH) Th displams masurd blow h quilibrium posiio ar posiiv. B dividig wih m, h las quaio a b wri as: d k d m or d ω d k Whrω. This quaio is kow as h quaio of simpl harmoi m moio or as h fr u-dampd moio...5 Iiial Codiios Assoiad wih h diffrial quaio d ω d ar h obvious iiial odiios ( ) α, ( ) β Ths iiial odiios rprs h iiial displam ad h iiial vloi. For ampl If α >, β < h h bod sars from a poi blow h quilibrium posiio wih a impard upward vloi. If α <, β h h bod sars from rs α uis abov h quilibrium posiio...6 Soluio ad Equaio of Moio Cosidr h quaio of simpl harmoi moio d ω d d m m Pu, m d Th h auiliar quaio is m ω m ± iω Thus h auiliar quaio has ompl roos. m ωi, m ωi H, h gral soluio of h quaio of simpl harmoi moio is osω siω ( ) Coprigh Virual Uivrsi of Pakisa 88

200 Diffrial Equaios (MTH)..7 Alraiv form of Soluio I is of ovi o wri h abov soluio i a alraiv simplr form. Cosidr ad suppos ha Th So ha ( ) osω siω A, φ R suh ha Asiφ, Aosφ A, aφ ( ) si ω osφ os ω si A B or ( ) Asi ( ω φ ) Th umbr φ is alld h phas agl; No ha: This form of h soluio of h quaio of simpl harmoi moio is vr usful baus Ampliud of fr vibraios boms vr obvious Th ims wh h bod rosss quilibrium posiio ar giv b si ( ω φ ) or ω φ π Whr is a o-gaiv igr. Th Naur of Simpl Harmoi Moio..8 Ampliud W kow ha h soluio of h quaio of simpl harmoi moio a b wri as ( ) si ( ω φ ) A Clarl, h maimum disa ha h suspdd bod a ravl o ihr sid of h quilibrium posiio is A. This maimum disa alld h ampliud of moio ad is giv b Ampliud A..9 A Vibraio or a Cl I ravllig from A o - A ad h bak o A, h vibraig bod ompls o vibraio or o l. φ Coprigh Virual Uivrsi of Pakisa 89

201 Diffrial Equaios (MTH)..Priod of Vibraio Th simpl harmoi moio of h suspdd bod is priodi ad i rpas is posiio afr a spifi im priodt. W kow ha h disa of h mass a a im is giv b Si A Asi si ( ω φ) π ω φ ω ( ω φ π) Asi ( ω φ ) Asi Thrfor, h disas of h suspdd bod from h quilibrium posiio a h ims π ad ar sam ω Furhr, vloi of h bod a a im is giv b d Aωos ω φ d ( ) π Aωos ω φ ω [ φ ] Aωos ω π ( ) Aωos ω φ Thrfor h vloi of h bod rmais ualrd if is irasd b π / ω. H h im priod of fr vibraios dsribd b h d ordr diffrial quaio d ω d is giv b π T ω..frqu Th umbr of vibraio /l ompld i a ui of im is kow as frqu of h fr vibraios, dod b f. Si h ls ompld i im T is. Thrfor, h umbr of ls ompld i a ui of im is / T H ω f T π Coprigh Virual Uivrsi of Pakisa 9

202 Diffrial Equaios (MTH) Eampl Solv ad irpr h iiial valu problm Irpraio Comparig h iiial odiios Wih W s ha d 6 d ( ), ( ). ( ), ( ). ( ) α, ( ) β α, β Thus h problm is quival o Pullig h mass o a sprig uis blow h quilibrium posiio. Holdig i hr uil im ad h rlasig h mass from rs. Soluio Cosidr h diffrial quaio Pu Th, h auiliar quaio is d 6 d m d, m d m 6 m ± i Thrfor, h gral soluio is: os si Now w appl h iiial odiios. Thus m ( ) ( ).. So ha ( ) os si Coprigh Virual Uivrsi of Pakisa 9

203 Diffrial Equaios (MTH) d d si os Thrfor ( ) ( ). Thus H, h soluio of h iiial valu problm is ( ) os No ha Clarl, h soluio shows ha o h ssm is s io moio, i sas i moio wih mass bouig bak ad forh wih ampliud big uis. Siω. Thrfor, h priod of osillaio is π π T sods Eampl A mass wighig lbs srhs a sprig 6 ihs. A h mass is rlasd from a poi 8 ihs blow h quilibrium posiio wih a upward vloi of f / s. Drmi h fuio () ha dsribs h subsqu fr moio. Soluio For osis of uis wih h girig ssm, w mak h followig ovrsios 6 ihs foo 8 ihs foo. Furhr wigh of h bod is giv o b W lbs Bu W mg Thrfor W m g or m slugs. 6 Si Srh s foo Thrfor b Hook s Law, w a wri Coprigh Virual Uivrsi of Pakisa 9

204 Diffrial Equaios (MTH) k k lbs/f H h quaio of simpl harmoi moio boms d m d 6 d d k d or 6. d Si h iiial displam is 8 ihs f ad h iiial vloi is f/s, h iiial odiios ar: ( ), ( ) Th gaiv sig idias ha h iiial vloi is giv i h upward i. gaiv dirio. Thus, w d o solv h iiial valu problm. Solv d 6 d Subj o ( ), ( ) m d Puig, m d W obai h auiliar quaio m 6 or m ± 8i Th gral soluio of h quaio is ( ) os8 si 8 Now, w appl h iiial odiios. ( ).. m Coprigh Virual Uivrsi of Pakisa 9

205 Diffrial Equaios (MTH) Thus So ha ( ) os8 si 8 Si 6 ( ) si 8 8 os8. Thrfor 6 ( ). 8. Thus. 6 H, soluio of h iiial valu problm is ( ) os8 si 8. 6 Eampl Wri h soluio of h iiial valu problm disussd i h prvious ampl i h form ( ) si ( ω φ ) A. Soluio Th iiial valu disussd i h prvious ampl is: d Solv 6 d Subj o ( ), ( ) Soluio of h problm is 8 6 Thus ampliud of moio is giv b ( ) os8 si A f Coprigh Virual Uivrsi of Pakisa 9

206 Diffrial Equaios (MTH) ad h phas agl is dfid b / si φ > 7 / 6 7 / 6 os φ < 7 / 6 7 Thrfor aφ or ( ).6 radias a Si si φ >, osφ <, h phas agl φ mus b i d quadra. Thus φ π.6.86 radias H h rquird form of h soluio is 7 ( ) si( 8.86) 6 Eampl For h moio dsribd b h iiial valu problm Solv d 6 d, Subj o ( ) ( ) Fid h firs valu of im for whih h mass passs hrough h quilibrium posiio hadig dowward. Soluio W kow ha h soluio of iiial valu problm is ( ) os8 si 8. 6 This soluio a b wri i h form 7 6 ( ) si( 8.86) Th valus of for whih h mass passs hrough h quilibrium posiio i. for whih ar giv b w φ π Coprigh Virual Uivrsi of Pakisa 95

207 Diffrial Equaios (MTH) Whr,,, hrfor, w hav 8.86 π, 8.86 π, 8.86 π,or. 66,. 558,. 95, H, h mass passs hrough h quilibrium posiio hadig dowward firs im a. 558 sods.. Eris Sa i words a possibl phsial irpraio of h giv iiial-valu problms.., ( ), ( ) 6., ( ).7, ( ) Wri h soluio of h giv iiial-valu problm i h form ( ) Asi ( ϖ φ ). 5, ( ), ( ). 8, ( ), ( ) 5., ( ), ( ) 6,, 6. ( ) ( ) 6.,, 7. ( ) ( ),, 8. ( ) ( ) 9. Th priod of fr udampd osillaios of a mass o a sprig is π / sods. If h sprig osa is 6 lb/f, wha is h umrial valu of h wigh?. A -lb wigh is aahd o a sprig, whos sprig osa is 6 lb/f. Wha is priod of simpl harmoi moio?. A -lb wigh, aahd o h sprig, srhs i ihs. Fid h quaio of h moio if h wigh is rlasd from rs from a poi ihs abov h quilibrium posiio.. A -lb wigh srhs a sprig 6 ihs. Th wigh is rlasd from rs 6 ihs blow h quilibrium posiio. π π π π 9π a) Fid h posiio of h wigh a,,,, sods. 8 6 b) Wha is h vloi of h wigh wh π / 6 sods? I whih dirio is h wigh hadig a his isa? ) A wha ims dos h wigh pass hrough h quilibrium posiio? Coprigh Virual Uivrsi of Pakisa 96

208 Diffrial Equaios (MTH) Dampd Moio I h prvious lur, w disussd h fr harmoi moio ha assums o rardig fors aig o h movig mass. Howvr No rardig fors aig o h movig bod is o ralisi, baus Thr alwas iss a las a rsisig for du o surroudig mdium. For ampl a mass a b suspdd i a visous mdium. H, h dampig fors d o b iludd i a ralisi aalsis.. Dampig For I h sud of mhais, h dampig fors aig o a bod ar osidrd o b d proporioal o a powr of h isaaous vloi. I h hdro damial d problms, h dampig for is proporioal o ( d / d). So ha i hs problms Dampig fo r d -β d Whr β is a posiiv dampig osa ad gaiv sig idias ha h dampig for as i a dirio opposi o h dirio of moio. I h prs disussio, w shall assum ha h dampig for is proporioal o h d isaaous vloi. Thus for us d Dampig fo r. Th Diffrial Equaio d -β d Suppos Tha A bod of mass m is aahd o a sprig. Th sprig srhs b a amou s o aai h quilibrium posiio. Th mass is furhr displad b a amou ad h rlasd. No ral fors ar imprssd o h ssm. Thrfor, hr ar hr fors aig o h mass, aml: a) Wigh mg of h bod b) Rsorig for k ( s ) d ) Dampig for -β d Thrfor, oal for aig o h mass m is Coprigh Virual Uivrsi of Pakisa 97

209 Diffrial Equaios (MTH) mg k d d ( s ) β So ha b Nwo s sod law of moio, w hav d mg k d m Si i h quilibrium posiio Thrfor mg ks d d ( s ) β d d k β d d m Dividig wih m, w obai h diffrial quaio of fr dampd moio d β d k d m d m For algbrai ovi, w suppos ha Th h quaio boms: β λ, ω m k m d d λ ω d d.. Soluio of h Diffrial Equaio Cosidr h quaio of h fr dampd moio Pu d d λ ω d d m, Th h auiliar quaio is: m d m d m, m m d d λm ω Solvig b us of quadrai formula, w obai m λ± λ ω Thus h roos of h auiliar quaio ar m λ λ, m λ λ ω ω Dpdig upo h sig of h quai λ ω, w a ow disiguish hr possibl ass of h roos of h auiliar quaio. Coprigh Virual Uivrsi of Pakisa 98

210 Diffrial Equaios (MTH) Cas Ral ad disi roos If λ ω > h β > k ad h ssm is said o b ovr-dampd. Th soluio of h quaio of fr dampd moio is m m ( ) λ λ ω λ ω or ( ) [ ] This quaio rprss smooh ad o osillaor moio. Cas Ral ad qual roos If λ ω, h β k ad h ssm is said o b riiall dampd, baus a sligh dras i h dampig for would rsul i osillaor moio. Th gral soluio of h diffrial quaio of fr dampd for is m m ( ) or ( ) λ ( ) Cas Compl roos If λ w <, h β < k ad h ssm is said o b udr-dampd. W d o rwri h roos of h auiliar quaio as: m λ ω λ i, m λ ω λ i Thus, h gral soluio of h quaio of fr dampd moio is ( ) λ λ λ os ω si ω This rprss a osillaor moio; bu ampliud of vibraio as baus of h offii λ. No ha Eah of h hr soluios oai h dampig faor, λ >, h displams of h mass bom gligibl for largr ims. λ Coprigh Virual Uivrsi of Pakisa 99

211 Diffrial Equaios (MTH).. Alraiv form of h Soluio Wh λ ω <, h soluio of h diffrial quaio of fr dampd moio is ( ) d d λ ω d d λ λ λ os ω si ω Suppos ha A ad φ ar wo ral umbrs suh ha So ha si φ, osφ A A A, aφ Th umbr φ is kow as h phas agl. Th h soluio of h quaio boms: ( ) A λ si ω λ osφ os λ or ( ) A si( ω λ φ ) No ha Th offii A λ ω λ siφ is alld h dampd ampliud of vibraios. Th im irval bw wo sussiv maima of ( ) ad is giv b h umbr π ω λ Th followig umbr is kow as h quasi frqu. ω λ π Th graph of h soluio λ ( ) A si ( ω λ φ ) rosss posiiv -ais, i. h li, a ims ha ar giv b ω λ φ π Whr,,,. For ampl, if w hav ( ).5 π si is alld quasi priod, Coprigh Virual Uivrsi of Pakisa

212 Diffrial Equaios (MTH) Th π π π π π or, π, π, π π 7π or,,, W oi ha diffr bw wo sussiv roos is π k k quasi priod π Si quasi priod π. Thrfor π k k quasi priod λ Si ( ) A wh si ω λ φ, h graph of h soluio λ ( ) A si( ω λ φ ) ouhs h graphs of h poial fuios ± A λ a h valus of for whih si( ω λ φ ) ± This mas hos valus of for whih or Agai, if w osidr ( ) π ω λ φ ( ) ( π / ) φ ω λ.5 ( ) si π whr,,,, Th * π π * π π * π 5π,,, Or * 5 π * π 7,, π, Agai, w oi ha h diffr bw sussiv valus is * * π k k Th valus of for whih h graph of h soluio λ ( ) A si( ω λ φ ) ouhs h poial graph ar o h valus for whih h fuio aais is rlaiv rmum. Coprigh Virual Uivrsi of Pakisa

213 Diffrial Equaios (MTH) Eampl Irpr ad solv h iiial valu problm d d 5 d d ( ), ( ) Fid rm valus of h soluio ad hk whhr h graph rosss h quilibrium posiio. Irpraio Comparig h giv diffrial quaio d d 5 d d wih h gral quaio of h fr dampd moio d d λ ω d d w s ha λ 5, ω so ha λ ω > Thrfor, h problm rprss h ovr-dampd moio of a mass o a sprig. Ispio of h boudar odiios ( ), ( ) rvals ha h mass sars ui blow h quilibrium posiio wih a dowward vloi of f/s. Soluio To solv h diffrial quaio d d 5 d d m d m d W pu, m, m d d Th h auiliar quaio is m 5m m m ( )( ) m Coprigh Virual Uivrsi of Pakisa

214 Diffrial Equaios (MTH) m, m, Thrfor, h auiliar quaio has disi ral roos m, m Thus h soluio of h diffrial quaio is: ( ) So ha ( ) Now, w appl h boudar odiios Thus ( ).. ( ) Solvig hs wo quaios, w hav. 5, Thrfor, soluio of h iiial valu problm is Ermum 5 ( ) Si ( ) Thrfor 5 d 5 8 d So ha ( ) or or. 57 Si l 8 5 d 5 d Thrfor a.57, w hav Coprigh Virual Uivrsi of Pakisa

215 Diffrial Equaios (MTH) So ha h soluio ( ) d d < has a maimum a. 57 ad maimum valu of is: (.57). 69 H h mass aais a rm displam of posiio. Chk Suppos ha h graph of ( ).69 f blow h quilibrium dos ross h ais, ha is, h mass passs hrough h quilibrium posiio. Th a valu of iss for whih ( ) 5 i. or l This valu of is phsiall irrlva baus im a vr b gaiv. H, h mass vr passs hrough h quilibrium posiio. Eampl A 8-lb wigh srhs a sprig f. Assumig ha a dampig for umriall quals o wo ims h isaaous vloi as o h ssm. Drmi h quaio of moio if h wigh is rlasd from h quilibrium posiio wih a upward vloi of f / s. Soluio Si Thrfor, b Hook s law Si Thrfor β Wigh 8 lbs, Srh s f 8 k k lb / f d Dampig for d Coprigh Virual Uivrsi of Pakisa

216 Diffrial Equaios (MTH) Also Wigh mass m g 8 slugs Thus, h diffrial quaio of moio of h fr dampd moio is giv b or m d d k β d d d d d d d d or 8 6 d d Si h mass is rlasd from quilibrium posiio wih a upward vloi f / s. Thrfor h iiial odiios ar:, ( ) ( ) Thus w d o solv h iiial valu problm d d Solv 8 6 d d Subj o ( ), ( ) Pu Thus h auiliar quaio is m m, d d m 8m 6 m, d d m m or ( ) m m, So ha roos of h auiliar quaio ar ral ad qual. m m H h ssm is riiall dampd ad h soluio of h govrig diffrial quaio is ( ) Morovr, h ssm is riiall dampd. W ow appl h boudar odiios. ( ).. Coprigh Virual Uivrsi of Pakisa 5

217 Diffrial Equaios (MTH) Thus ( ) d d So ha ( ). Thus soluio of h iiial valu problm is Ermum ( ) Si ( ) Thrfor Thus d d ( ) d d Th orrspodig rm displam is Thus h wigh rahs a maimum high of Eampl.76 f.76 f abov h quilibrium posiio. A 6-lb wigh is aahd o a 5 - f log sprig. A quilibrium h sprig masurs 8.f.If h wigh is pushd up ad rlasd from rs a a poi - f abov h quilibrium posiio. Fid h displam ( ) if i is furhr kow ha h surroudig mdium offrs a rsisa umriall qual o h isaaous vloi. Soluio Lgh of u - srhd sprig 5 f Lgh of sprig a quilibrium 8. f Thus Elogaio of sprig s B Hook s law, w hav Furhr. f (.) 5 lb / f 6 k k Wigh mass m g 6 slugs Coprigh Virual Uivrsi of Pakisa 6

218 Diffrial Equaios (MTH) Si Dampig for Thrfor β Thus h diffrial quaio of h fr dampd moio is giv b or d m k d d 5 d d β d d d d d or d d Si h sprig is rlasd from rs a a poi f abov h quilibrium posiio. Th iiial odiios ar: ( ), ( ) H w d o solv h iiial valu problm d d d d ( ), ( ) To solv h diffrial quaio, w pu Th h auiliar quaio is, d m d, d d m m d m d m m or m ± i So ha h auiliar quaio has ompl roos m i, m i Th ssm is udr-dampd ad h soluio of h diffrial quaio is: ( ) ( si ) os Now w appl h boudar odiios.. ( ) m. Coprigh Virual Uivrsi of Pakisa 7

219 Diffrial Equaios (MTH) Thus ( ) ( os si ) ( si os ) ( os si ) d d Thrfor ( ) 6 H, soluio of h iiial valu problm is Eampl ( ) os si Wri h soluio of h iiial valu problm i h alraiv form d d d d, ( ) ( ) ( ) A si ( φ ) Soluio W kow from prvious ampl ha h soluio of h iiial valu problm is Suppos ha ( ) os si A ad φ ar ral umbrs suh ha / siφ, osφ A A Th A 9 Also a φ / Thrfor ( ).9radia a Si si φ <, osφ <, h phas agl φ mus b i rd quadra. Thrfor φ π.9.9 radias Coprigh Virual Uivrsi of Pakisa 8

220 Diffrial Equaios (MTH) H Th valus of valus γ ( ) si(.9) whr h graph of h soluio rosss posiiv - ais ad h * γ whr h graph of h soluio ouhs h graphs of giv i h followig abl... Quasi Priod γ γ Si ( ) si(.9) So ha h quasi priod is giv b * ( ) * γ γ λ ω λ π ω π sods ± ar H, diffr bw h sussiv * π γ ad γ is uis.. Eris Giv a possibl irpraio of h giv iiial valu problms.., ( ), ( ) , ( ), ( ). A -lb wigh is aahd o a sprig whos osa is lb /f. Th mdium offrs a rsisa o h moio of h wigh umriall qual o h isaaous vloi. If h wigh is rlasd from a poi f abov h quilibrium posiio wih a dowward vloi of 8 f / s, drmi h im ha h wigh passs hrough h quilibrium posiio. Fid h im for whih h wigh aais is rm displam from h quilibrium posiio. Wha is h posiio of h wigh a his isa?. A -f sprig masurs 8 f log afr a 8-lb wigh is aahd o i. Th mdium hrough whih h wigh movs offrs a rsisa umriall qual o ims Coprigh Virual Uivrsi of Pakisa 9

221 Diffrial Equaios (MTH) h isaaous vloi. Fid h quaio of moio if h wigh is rlasd from h quilibrium posiio wih a dowward vloi of 5 f / s. Fid h im for whih h wigh aais is rm displam from h quilibrium posiio. Wha is h posiio of h wigh a his isa? 5. A -kg mass is aahd o a sprig whos osa is 6 N / m ad h ir ssm is h submrgd i o a liquid ha impars a dampig for umriall qual o ims h isaaous vloi. Drmi h quaios of moio if a. Th wigh is rlasd from rs m blow h quilibrium posiio; ad b. Th wigh is rlasd m blow h quilibrium posiio wih ad upward vloi of m/s. 6. A for of -lb srhs a sprig f. A.-lb wigh is aahd o h sprig ad h ssm is h immrsd i a mdium ha impars dampig for umriall qual o. ims h isaaous vloi. a. Fid h quaio of moio if h wigh is rlasd from rs f abov h quilibrium posiio. b. Eprss h quaio of moio i h form λ ( ) A si ( ) ω λ φ. Fid h firs ims for whih h wigh passs hrough h quilibrium posiio hadig upward. 7. Afr a -lb wigh is aahd o a 5-f sprig, h sprig masurs 7-f log. Th -lb wigh is rmovd ad rplad wih a 8-lb wigh ad h ir ssm is plad i a mdium offrig a rsisa umriall qual o h isaaous vloi. a. Fid h quaio of moio if h wigh is rlasd /f blow h quilibrium posiio wih a dowward vloi of f / s. b. Eprss h quaio of moio i h form λ ( ) A si ( ) ω λ φ. Fid h im for whih h wigh passs hrough h quilibrium posiio hadig dowward. 8. A -lb wigh aahd o a sprig srhs i f. Th wigh is aahd o a dashpo-dampig dvi ha offrs a rsisa umriall qual o β ( β > ) ims h isaaous vloi. Drmi h valus of h dampig osa β so ha h subsqu moio is a. Ovr-dampd b. Criiall dampd. Udr-dampd 9. A mass of g. srhs a sprig m. A dampig dvi impars a rsisa o moio umriall qual o 56 (masurd i ds /(m / s)) ims h isaaous vloi. Fid h quaio of moio if h mass is rlasd from h quilibrium posiio wih dowward vloi of m / s.. Th quasi priod of a udr-dampd, vibraig -slugs mass of a sprig isπ / sods. If h sprig osa is 5 lb / f, fid h dampig osa β. Coprigh Virual Uivrsi of Pakisa

222 Diffrial Equaios (MTH) Ford Moio I his las lur o h appliaios of sod ordr liar diffrial quaios, w osidr A vibraioal ssm osisig of a bod of mass m aahd o a sprig. Th moio of h bod is big driv b a ral for f ( ) i.. ford moio. Flow of urr i a lrial irui ha osiss of a iduor, rsisor ad a apaior od i sris, baus of is similari wih h ford moio.. Ford moio wih dampig Suppos ha w ow ak io osidraio a ral for f ( ) aig o h ssm ar: a) Wigh of h bod mg b) Th rsorig for k ( s ) ) Th dampig ff β ( d / d) d) Th ral for f ( ).. Th, h fors H dos h disa of h mass m from h quilibrium posiio. Thus h oal for aig o h mass m is giv b B h Nwo s d law of moio, w hav d For mg k β d d For ma m d d d ( s ) f ( ) Thrfor m mg ks k β f ( ) Bu mg ks So ha d d β d m d d d d d k m d d f ( ) m or λ ω F( ) whr F( ) No ha ( ) β f k, λ ad ω. m m m Coprigh Virual Uivrsi of Pakisa

223 Diffrial Equaios (MTH) Th las quaio is a o-homogous diffrial quaio govrig h ford moio wih dampig. To solv his quaio, w us ihr h mhod of udrmid offiis or h variaio of paramrs. Eampl Irpr ad solv h iiial valu problm d d. 5os 5 d d ( ), ( ) Irpraio Th problm rprss a vibraioal ssm osisig of A mass m slugs or kilograms 5 Th mass is aahd o a sprig havig sprig osa k lb / f or N / m Th mass is rlasd from rs f or mr blow h quilibrium posiio Th moio is dampd wih dampig osa β.. Th moio is big driv b a ral priodi for f ( ) 5os ha has π priodt. Soluio Giv h diffrial quaio d d. 5os 5 d d d d or 6 5os d d Firs osidr h assoiad homogous diffrial quaio. d d 6 d d m d m d Pu, m, m d d Th h auiliar quaio is: m 6m m Coprigh Virual Uivrsi of Pakisa

224 Diffrial Equaios (MTH) m ± Thus h auiliar quaio has ompl roos m i, m i So ha h omplmar fuio of h quaio is ( os si ) To fid a pariular igral of o-homogous diffrial quaio w us h udrmid offiis, w assum ha p Aos Bsi ( ) Th p ( ) Asi B os p ( ) 6Aos 6Bsi So ha i p 6 p p 6Aos 6Bsi Asi B os Aos Bsi ( 6A B) os ( A 6B) si Subsiuig i h giv o-homogous diffrial quaio, w obai ( 6 A B) os ( A 6B) si 5os Equaig offiis, w hav 6 A B 5 A 6B Solvig hs quaios, w obai 5 5 A, B p os si 5 Thus ( ) H h gral soluio of h diffrial quaio is: ( ) [ si ] os si os ( ) [ os si ] ( si os) si os Coprigh Virual Uivrsi of Pakisa

225 Diffrial Equaios (MTH) Now ( ) or or Also ( ) givs givs 5 86 or H h soluio of h iiial valu problm is: os si os si ( ). Trasi ad Sad-Sa Trms Du o h prs of h faor w oi ha h omplmar fuio ( ) 8 86 os si 5 5 posssss h propr ha lim ( ) Thus for larg im, h displams of h wigh ar losl approimad b h pariular soluio 5 5 p 5 ( ) os si Si ( ) as pariular soluio p ( ) is alld h sad-sa soluio, i is said o b rasi rm or rasi soluio. Th H, wh F is a priodi fuio, suh as Th gral soluio of h quaio F ( ) F γ or F( ) F osγ si Coprigh Virual Uivrsi of Pakisa

226 Diffrial Equaios (MTH) osiss of d d λ ω d d F( ) ( ) Trasi soluio Sad Sa Soluio Eampl Solv h iiial valu problm d d os si d d ( ), ( ) Soluio Firs osidr h assoiad homogous liar diffrial quaio d d d d m m m Pu, m, m Th h auiliar quaio is m m ± 8 or m ± i Thus h omplmar fuio is ( si ) os For h pariular igral w assum ha p Aos Bsi p p Asi B os Aos Bsi So ha Coprigh Virual Uivrsi of Pakisa 5

227 Diffrial Equaios (MTH) p d d d dp p Aos Bsi Asi B os Aos Bsi d p p or ( ) ( ) d d p A B os A B si d Subsiuig i h giv diffrial quaio, w hav ( A B) os ( A B) si os si Equaig offiis, w obai A B A B Solvig hs wo quaios, w hav: A, B Thus p si H gral soluio of h diffrial quaio is p or ( ) ( si ) si os os Thus ( ) ( si ) ( si os) os Now w appl h boudar odiios. ( ).. ( ). Thus soluio of h iiial valu problm is si si Si si as Thrfor si Trasi Trm, si Sad Sa Coprigh Virual Uivrsi of Pakisa 6

228 Diffrial Equaios (MTH) H si si Trasi Sadsa W oi ha h ff of h rasi rm boms gligibl for abou > π. Moio wihou Dampig If h ssm is imprssd upo b a priodi for ad hr is o dampig for h hr is o rasi rm i h soluio. Eampl Solv h iiial valu problm Whr F is a osa Soluio o d ω d Fo si γ ( ), ( ) For omplmar fuio, osidr h assoiad homogous diffrial quaio Pu d ω d Th h auiliar quaio is m m, m Thus h omplmar fuio is m ω m ± ωi ( ) ω si ω os To fid a pariular soluio, w assum ha p ( ) Aos γ Bsi γ Th ( ) Aγsi γ Bγ os γ p p ( ) Aγ os γ Bγ si γ Thrfor, p ω p Aγ os γ Bγ si γ Aω os γ Bω si γ Coprigh Virual Uivrsi of Pakisa 7

229 Diffrial Equaios (MTH) p ω p A ( ω γ ) osγ B( ω γ ) si γ Subsiuig i h giv diffrial quaio, w hav Equaig offiis, w hav ( ω γ ) os γ B( ω γ ) si γ F si A o γ ( ω γ ), B( ω γ ) Fo A Solvig hs wo quaios, w obai A ( γ ω ) F o, B ω γ Thrfor ( ) si γ p Fo ω γ H, h gral soluio of h diffrial quaio is Fo si ω γ F Th ( ) si os o γ ω ω ω ω os γ ω γ Now w appl h boudar odiios.. ( ) os ω si ω γ ( ) F γ. ω. ω γ o ( ) γf o ω( ω γ ) Thus soluio of h iiial valu problm is F ( ) o ( ) ( γ si ω ω si γ ), ( γ ω ) ω ω γ No ha h soluio is o dfid for ω lim a b obaid usig h L Hôpial s rul γ, Howvr ( ) γ ω Coprigh Virual Uivrsi of Pakisa 8

230 Diffrial Equaios (MTH) Clarl ( ) as ( ) lim F γ ω o d dγ F o lim ω γ ω F o F o γ si ω ωsi γ ( γ ) ω ω ( γ siω ω siγ) d d ω ω γ ( γ ) si ω ω os γ lim ω γ ω ωγ si ω ω osω ω Fo Fo si ω osω ω ω.thrfor hr is o rasi rm wh hr is o dampig for i h prs of a priodi imprssd for.. Elri Ciruis Ma diffr phsial ssms a b dsribd b a sod ordr liar diffrial quaio similar o h diffrial quaio of h ford moio: d d m β k f ( ) d d O suh aalogous as is ha of a LRC-Sris irui. Baus of h similari i mahmais ha govrs hs wo ssms, i migh b possibl o us our iuiiv udrsadig of o o hlp udrsad h ohr..5 Th LRC Sris Ciruis Th LRC sris irui osis of a iduor, rsisor ad apaior od i sris wih a im varig sour volag E (),.5. Rsisor A rsisor is a lrial ompo ha limis or rgulas h flow of lrial urr i a lrial irui. Th masur of h o whih a rsisor impds or rsiss wih h flow of urr hrough i is alld rsisa, dod b R. Clarl highr h rsisa, lowr h flow of urr. Lowr h rsisa, highr h flow of urr. Thrfor, w olud ha h flow of urr is ivrsl proporioal o h rsisa, i. Coprigh Virual Uivrsi of Pakisa 9

231 Diffrial Equaios (MTH) I V. R V IR WhrV is osa of proporioali ad i rprss h volag. Th abov quaio is mahmaial sam of h wll kow as Ohm s Law..5. Iduor A iduor is a passiv lroi ompo ha sors rg i h form of magi fild. I is simpls form h oduor osiss of a wir loop or oil woud o som suiabl marial. Whvr urr hrough a iduor hags, i. irass or drass, a our mf is idud i i, whih ds o oppos his hag. This propr of h oil du o whih i opposs a hag of urr hrough i is alld h idua. di Suppos ha I dos h urr h h ra of hag of urr is giv b This d di produs a our mf volagv. Th V is dirl proporioal o d di Vα d di V L d Whr L is osa of proporioali, whih rprss idua of h iduor. Th sadard ui for masurm of idua is Hr, dod b H..5. Capaior A apaior is a passiv lroi ompo of a lroi irui ha has h abili o sor harg ad opposs a hag of volag i h irui. Th abili of a apaior o sor harg is alld apaia of h apaior dod bc. If q oulomb of a harg o h apaior ad h poial diffr of V vols is sablishd bw plas of h apaior h q α C q CV or V q C WhrC is alld osa of proporioali, whih rprs apaia. Th sadard ui o masur apaia is farad, dod b F..6 Kirhhoff s Volag Law Th Kirhhoff s d law sas ha h sum of h volag drops aroud a losd loop quals h sum of h volag riss aroud ha loop. I ohr words h algbrai sum of volags aroud h los loop is zro. Coprigh Virual Uivrsi of Pakisa

232 Diffrial Equaios (MTH).6. Th Diffrial Equaio Now w osidr h followig irui osisig of a iduor, a rsisor ad a E. apaior i sris wih a im varig volag sour ( ) If VL, VR adv do h volag drop aross h iduor, rsisor ad apaior rspivl. Th di VL L, VR RI, V d Now b Kirhhoff s law, h sum of V V adv VL VR V di d q C L R q C, mus qual h sour volag E( ) E( ) or L RI E( ) dq Si h lri urr I rprss h ra of flow of harg. Thrfor, w a d wri dq I d Subsiuig i h las quaio, w hav: d q dq q L R d d C E( ) No ha: W hav s his quaio bfor! I is mahmaiall al h sam as h quaio for a driv, dampd harmoi osillaor. If ( ), R E h lri vibraio of h irui ar said o b fr dampd osillaio. If E( ), R h h lri vibraio a b alld fr u-dampd osillaios..6. Soluio of h diffrial quaio Th diffrial quaio ha govrs h flow of harg i a LRC-Sris irui is i. Coprigh Virual Uivrsi of Pakisa

233 Diffrial Equaios (MTH) d q dq q L R d d C E( ) This is a o-homogous liar diffrial quaio of ordr-. Thrfor, h gral soluio of his quaio osiss of a omplmar fuio ad pariular igral. For h omplmar fuio w fid gral soluio of h assoiad homogous diffrial quaio d q dq L R d d q C m dq m W pu q, m, d d q m d Th h auiliar quaio of h assoiad homogous diffrial quaio is: Lm Rm C If R h, dpdig o h disrimia, h auiliar quaio ma hav Ral ad disi roos Ral ad qual roos Compl roos Cas Ral ad disi roos L If Dis R > C Th h auiliar quaio has ral ad disi roos. I his as, h irui is said o b ovr dampd. Cas Ral ad qual L If Dis R Th h auiliar quaio has ral ad qual roos. I his as, h irui is said o b riiall dampd. Cas Compl roos If Dis R L < Th h auiliar quaio has ompl roos. I his as, h irui is said o b udr dampd. No ha m Coprigh Virual Uivrsi of Pakisa

234 Diffrial Equaios (MTH) Si b h quadrai formula, w kow ha R ± R L / m L I ah of h abov miod hr ass, h gral soluio of h ohomogous govrig quaio oais h faor. R / L Thrfor q ( ) as q q h harg o h apaior osillas as i das. This mas ha h apaior is hargig ad dishargig as E, ad R, h lrial vibraio do o approah zro as. This mas ha h rspos of h irui is Simpl Harmoi. I h udr dampd as wh ( ) o I h udr dampd as, i.. wh ( ) Coprigh Virual Uivrsi of Pakisa

235 Diffrial Equaios (MTH) 5 Ford Moio (Eampls) Eampl Cosidr a LC sris irui i whih E ( ) Drmi h harg q( ) o h apaior for > iiiall hr is o urr flowig i h irui. Soluio Si i a LC sris irui, hr is o rsisor. Thrfor, dq R d So ha, h govrig diffrial quaio boms d q L d Th iiial odiios for h irui ar q ( ) q, I( ) q o dq d Si I( ) Thrfor h iiial odiios ar quival o ( ) q, q ( ) q o Thus, w hav o solv h iiial valu problm. d q L d q ( ) q, q ( ) q o To solv h govrig diffrial quaio, w pu So ha h auiliar quaio is: q m, Lm m m ± L d q m d i L Thrfor, h soluio of h diffrial quaio is : m if is iiial harg is q o ad if Coprigh Virual Uivrsi of Pakisa

236 Diffrial Equaios (MTH) q os L ( ) si Now, w appl h boudar odiios q o. q ( ) qo qo L. q qo os si L L Diffriaig w.r o, w hav: Thus ( ) Now ( ) H Si ( ) dq qo si os d L L L L q. L q L ( ) q os I o dq d Thrfor, urr i h irui is giv b Eampl q I o L LC ( ) si Fid h harg ( ) Ohms, C. farad, E ( ), q ( ) qo ad ( ) q o h apaior i a LRC sris irui wh L.5 Hr, R I. Soluio W kow ha for a LRC irui, h govrig diffrial quaio is d q dq q L R d d Si L.5, R,C. Thrfor, h quaio boms: E( ) Coprigh Virual Uivrsi of Pakisa 5

237 Diffrial Equaios (MTH) d q dq q d d d q dq or q d d Th iiial odiios ar ( ) q, I( ) q o or ( ) q, ( ) q o q To solv h diffrial quaio, w pu q m Thrfor, h auiliar quaio is m dq m d q, m, m d d m ± 6 6 m m ± 6 i Thus, h soluio of h diffrial quaio is Now, w appl h iiial odiios q ( ) ( 6 si 6) os ( ) qo.. qo q q o Thrfor q( ) ( q 6 si 6) o os Now q ( ) ( q os6 os6) ( 6q si 6 6 os6) o o q q 6. Thus ( ) o m q o H h soluio of h iiial valu problm is ( ) q q o os 6 si 6 As disussd i h prvious lurs, a sigl si fuio Coprigh Virual Uivrsi of Pakisa 6

238 Diffrial Equaios (MTH) Si q R ad lim q ( ) q o ( ) si( 6.9) Thrfor h soluio of h giv diffrial quaio is rasi soluio. No ha Th lri vibraios i his as ar fr dampd osillaios as hr is o imprssd E o h irui. volag ( ) Eampl Fid h sad sa of soluio q p ( ) ad h sad sa urr i a LRC sris irui wh h imprssd volag is E( ) E siγ Soluio o Th sad sa soluio q p ( ) is a pariular soluio of h diffrial quaio d q dq L R q E si o γ d d C W us h mhod of udrmid offiis, for fidig q p ( ). Thrfor, w assum ( ) siγ os q A B γ q Aγ γ Bγ γ Th ( ) os si Thrfor d q L d dq R d q ( ) Aγ siγ Bγ osγ q ALγ siγ BLγ osγ ARγ osγ C A B BRγ siγ siγ osγ C C A ALγ C B BRγ siγ BLγ C Subsiuig i h giv diffrial quaio, w obai A ALγ C Equaig offiis of B BRγ siγ BLγ C si γ ad osγ, w obai A AL C γ BRγ E o ARγ osγ E ARγ osγ o siγ Coprigh Virual Uivrsi of Pakisa 7

239 Diffrial Equaios (MTH) B BL AR C γ γ or o E BR A L C γ γ B L C AR γ γ To solv hs quaios, w hav from sod quaio AR B L C γ γ Subsiuig i h firs quaio ad simplifig, w obai o E L C A L L R C C γ γ γ γ γ Usig his valu of A ad simplifig ilds R C C L L R E B o γ γ γ If w us h oaios h γ γ γ γ C C L L X C L X h R C C L L Z R X Z γ γ Th B, Z R E Z X E A o o γ γ Thrfor, h sad-sa harg is giv b ( ) Z R E Z X E q o o p γ γ γ γ os si So ha h sad-sa urr is giv b ( ) Z X Z R Z E I o p γ γ os si No ha Th quai γ γ C L X is alld h raa of h irui. Th quai R X Z is alld impda of h irui. Boh h raa ad h impda ar masurd i ohms. Coprigh Virual Uivrsi of Pakisa 8

240 Diffrial Equaios (MTH) Eris. A 6-lb wigh srhs a sprig 8/ f. Iiiall h wigh sars from rs -f blow h quilibrium posiio ad h subsqu moio aks pla i a mdium ha offrs a dampig for umriall qual o ½ h isaaous vloi. Fid h quaio of moio, if h wigh is driv b a ral for qual o f ( ) os.. A mass -slug, wh aahd o a sprig, srhs i -f ad h oms o rs i h quilibrium posiio. Sarig a, a ral for qual o ( ) 8si f is applid o h ssm. Fid h quaio of moio if h surroudig mdium offrs a dampig for umriall qual o 8 ims h isaaous vloi.. I problm drmi h quaio of moio if h ral for is ( ) si f. Aalz h displams for.. Wh a mass of kilograms is aahd o a sprig whos osa is N/m, i oms o rs i h quilibrium posiio. Sarig a, a for qual o ( ) 68 os f is applid o h ssm. Fid h quaio of moio i h abs of dampig. 5. I problm wri h quaio of moio i h form ( ) Asi( ω φ) B si( θ ). Wha is h ampliud of vibraios afr a vr log im? 6. Fid h harg o h apaior ad h urr i a LC sris irui. Whr L Hr, C farad, E( ) 6 vols. Assumig ha q ( ) ad i( ) Drmi whhr a LRC sris irui, whr L Hrs, R ohms, C.farad is ovr-dampd, riiall dampd or udr-dampd. 8. Fid h harg o h apaior i a LRC sris irui wh L / Hr, R ohms, C / farad, E vols, q oulombs ad i Is h harg o h apaior vr qual o zro? Fid h harg o h apaior ad h urr i h giv LRC sris irui. Fid h maimum harg o h apaior. ( ) ( ) ( ) amprs 9. L R C E( ) q( ) i ( ) amprs. L hr, R ohms, C. farad, E( ) vols, q( ) i ( ) amprs 5/ hrs, ohms, / farad, vols, oulombs, oulombs, Coprigh Virual Uivrsi of Pakisa 9

241 Diffrial Equaios (MTH) 6 Diffrial Equaios wih Variabl Coffiis So far w hav b solvig Liar Diffrial Equaios wih osa offiis. W will ow disuss h Diffrial Equaios wih o-osa (variabl) offiis.ths quaios ormall aris i appliaios suh as mpraur or poial u i h rgio boudd bw wo ori sphrs. Th udr som d u du irumsas w hav o solv h diffrial quaio r dr dr whr h variabl r> rprss h radial disa masurd ouward from h r of h sphrs. Diffrial quaios wih variabl offiis suh as ( v ) ( ) ( ) ad our i appliaios ragig from poial problms, mpraur disribuios ad vibraio phoma o quaum mhais. Th diffrial quaios wih variabl offiis ao b solvd so asil. 6. Cauh- Eulr Equaio A liar diffrial quaio of h form d d d a a a a g( ) d d d whr a, a,, a ar osas, is said o b a Cauh-Eulr quaio or quidimsioal quaio. Th dgr of ah moomial offii mahs h ordr of diffriaio i. is h offii of h drivaiv of, of (-)h drivaiv of,. For ovi w osidr a homogous sod-ordr diffrial quaio d d a b, d d Th soluio of highr-ordr quaios follows aalogousl. Also, w a solv h o-homogous quaio d d a b g( ), d d b variaio of paramrs afr fidig h omplmar fuio (). Coprigh Virual Uivrsi of Pakisa

242 Diffrial Equaios (MTH) W fid h gral soluio o h irval (, ) ad h soluio o (, ) a b obaid b subsiuig i h diffrial quaio. 6.. Mhod of Soluio W r a soluio of h form drivaivs ar, rspivl, d d m m ad m, whr m is o b drmid. Th firs ad sod d ( ) m mm d Cosqul h diffrial quaio boms d d d d m m a b a m( m ) b m ) m m am ( m bm m ( am( m ) bm ) m Thus is a soluio of h diffrial quaio whvr m is a soluio of h auiliar quaio ( am ( m ) bm ) or am ( b a) m Th soluio of h diffrial quaio dpds o h roos of h AE. 6.. Cas-I (Disi Ral Roos) L m ad m do h ral roos of h auiliar quaio suh ha m m. Th m m ad form a fudamal s of soluios. H h gral soluio is Eampl Solv Soluio: Suppos ha m m. d d d d m, h d d m m Now subsiuig i h diffrial quaio, w g: d d d d m m m( m ) m m ( m m ) if m m This implis m, m ; roos ar ral ad disi., m d ( ) m mm d m m m ( mm ( ) m ) Coprigh Virual Uivrsi of Pakisa

243 Diffrial Equaios (MTH) So h soluio is 6.. Cas II (Rpad Ral Roos). If h roos of h auiliar quaio ar rpad, ha is, h w obai ol o m soluio. To osru a sod soluio, w firs wri h Cauh-Eulr quaio i h form d b d d a d a d d Comparig wih P( ) Q( ) d d b W mak h idifiaio P ( ). Thus a b a d m d m ( ) b ( ) l a m d m b m a m. d Si roos of h AE am ( b a) m ar qual, hrfor disrimia is zro i. m ( b a) ( b a) or m a a b ba m a. a d m d m l. Th gral soluio is h m m l d d Eampl Solv 8. d d m d m Soluio: Suppos ha, h m d Subsiuig i h diffrial quaio, w g: d m, m( m ). d Coprigh Virual Uivrsi of Pakisa

244 Diffrial Equaios (MTH) d d m m 8 (m( m ) 8m ) (m m ) d d if m m or (m ). Si l. m, h gral soluio is For highr ordr quaios, if m is a roo of muliplii k, h i a b show ha: m, m l, m (l ),, m (l ) k ar k liarl idpd soluios. Corrspodigl, h gral soluio of h diffrial quaio mus h oai a liar ombiaio of hs k soluios. 6.. Cas III (Cojuga Compl Roos) If h roos of h auiliar quaio ar h ojuga pair m α i, m α iβ β whr α ad β > ar ral, h h soluio is α iβ i α β. Bu, as i h as of quaios wih osa offiis, wh h roos of h auiliar quaio ar ompl, w wish o wri h soluio i rms of ral fuios ol. W o h idi iβ ( l ) iβ iβl, whih, b Eulr s formula, is h sam as Similarl w hav iβ os( βl ) isi( βl ) iβ os( βl ) isi( βl ) Addig ad subraig las wo rsuls ilds, rspivl, i β i β os( β l ) ad i β i β isi( β l ) From h fa ha α iβ i α β is h soluio of a b, for a valus of osas ad, w s ha α ( iβ iβ), ( ) α ( iβ iβ), (, ) or α (os( β l )), α (si( β l )) ar also soluios. Coprigh Virual Uivrsi of Pakisa

245 Diffrial Equaios (MTH) α α α Si W ( os( β l ), si( β l )) β ; β >, o h irval (, ), w α α olud ha os( β l ) ad si( β l ) osiu a fudamal s of ral soluios of h diffrial quaio. H h gral soluio is α [ os( β l ) si( β l )] d d Eampl Solv h iiial valu problm, ( ), () 5 d d Soluio: L us suppos ha: m, h d d m m d d m m ( m( m ) m ) ( m m ) d d if m m. From h quadrai formula w fid ha h idifiaios α ad β d m ad m( m ). d m i ad m i. If w mak quaio is os( l ) si( l )]. [, so h gral soluio of h diffrial B applig h odiios ( ), () 5, w fid ha ad. Thus h soluio o h iiial valu problm is [os( l ) si( l )] Eampl Solv h hird-ordr Cauh-Eulr diffrial quaio Soluio d d d 5 7 8, d d d Th firs hr drivaiv of d d m m, d m( m ) d m ar m so h giv diffrial quaio boms, d m mm ( )( m ), d d d d m m m m( m )( m ) 5 m( m ) 7m 8 d d d m m ( m( m )( m ) 5m( m ) 7m 8) m ( m m m 8) m I his as w s ha is a soluio of h diffrial quaio, providd m is a roo of h ubi quaio m m m 8, Coprigh Virual Uivrsi of Pakisa

246 Diffrial Equaios (MTH) or ( m )( m ) Th roos ar: m, m i, m i. H h gral soluio is os( l ) si(l ) Eampl 5 Solv h o-homogous quaio m d m d m Soluio Pu m, m( m ) d d Thrfor w g h auiliar quaio, m ( m ) m or ( m )( m ) or m, Thus Bfor usig variaio of paramrs o fid h pariular soluio p u u, W W rall ha h formulas u ad u W W, whr W, W f ( ) f ( ), ad W is h Wroskia of ad, wr drivd udr h assumpio ha h diffrial quaio has b pu io spial form. P( ) Q( ) f ( ) Thrfor w divid h giv quaio b, ad form w mak h idifiaio f ( ). Now wih,, ad 5 W, W, W w fid u 5 ad ad u H p u u Fiall w hav ( 6. Eriss. ) u u. p Coprigh Virual Uivrsi of Pakisa 5

247 Diffrial Equaios (MTH) d d d d d d 7. d d 6 9 d d d d d d ; (), () 9. l. d d d 6 6 l d d d Coprigh Virual Uivrsi of Pakisa 6

248 Diffrial Equaios (MTH) 7 Cauh-Eulr Equaio (Alraiv Mhod of Soluio) W rdu a Cauh-Eulr diffrial quaio o a diffrial quaio wih osa offiis hrough h subsiuio or or or l d d d d d d d d d d d d d d d d d d ( ) ( ) d d d d d d d ( ) d d d d d d d Thrfor, d d Now irodu h oaio d d d d d d d d D, D,. d d d d ad,,. d d Thrfor, w hav D D ( ) d d d d Similarl D ( )( ) D ( )( )( ) so o so forh. This subsiuio i a giv Cauh-Eulr diffrial quaio will rdu i io a diffrial quaio wih osa offiis. d d Coprigh Virual Uivrsi of Pakisa 7

249 Diffrial Equaios (MTH) A his sag w suppos m o obai a auiliar quaio ad wri h soluio i rms of ad. W h go bak o hrough. Coprigh Virual Uivrsi of Pakisa 8

250 Diffrial Equaios (MTH) d d Eampl Solv d d Soluio Th giv diffrial quaio a b wri as ( D Wih h subsiuio D ) or l, w obai D, D ( ) Thrfor h quaio boms: [ ( ) ] or ( ) d d or d d Now subsiu: Thus m h d mm d, d m m d ( m m ) m or m m, whih is h auiliar quaio. ( m )( m ) m, Th roos of h auiliar quaio ar disi ad ral, so h soluio is Bu, hrfor h aswr will b d d Eampl Solv 8 d d Soluio Th diffrial quaio a b wri as: ( D 8D ) d d Whr D, D d d Now wih h subsiuio Th quaio boms: or l, D, D ( ) whr d d Coprigh Virual Uivrsi of Pakisa 9

251 Diffrial Equaios (MTH) ( ( ) 8 ) or ( ) d d d d Now subsiuig m h (m m ) m d mm d, d d m m, w g or m m or (m ) or m, ; h roos ar ral bu rpad. Thrfor h soluio is ( ) or i- ( l ) l d d Eampl Solv h iiial valu problm, ( ), () 5 d d Soluio Th giv diffrial a b wri as: ( D Now wih h subsiuio D ) D, D ( ) Thus h quaio boms: or l w hav: ( ( ) ) or ( ) Pu d d d d m h h A.E. quaio is: or m m ± or m ± i Coprigh Virual Uivrsi of Pakisa

252 Diffrial Equaios (MTH) So ha soluio is: ( os si ) or ( os l si l ) Now ( ) givs, ( os si ) ( os l si l ) ( si l os l ) ( ) 5 givs: 5 [ ] [ ] or 5, H soluio of h IVP is: [os( l ) si( l )]. d d d Eampl Solv d d d Soluio Th giv diffrial quaio a b wri as: ( D Now wih h subsiuio 5 D 7D 8) or l w hav: D, D ( ), D ( )( ) So h quaio boms: ( ( )( ) 5 ( ) 7 8) or ( ) or ( 8) Pu d d d or 8 d d d m, h h auiliar quaio is: m m m 8 or ( m )( m ) m, or ± i So h soluio is: os si Coprigh Virual Uivrsi of Pakisa

253 Diffrial Equaios (MTH) or os(l ) si(l ) Eampl 5 Solv h o-homogous diffrial quaio Soluio Firs osidr h assoiad homogous diffrial quaio. d d Wih h oaio d d ( D D ) D, D, h diffrial quaio boms: or l Wih h subsiuio, w hav: D, D ( ) So h homogous diffrial quaio boms: [ ( ) ] ( ) Pu d d or d d m h h AE is: m m or ( m )( m ), or m,, as For p w wri h diffrial quaio as: p u u, whr u ad u ar fuios giv b W W u, u W, W wih Coprigh Virual Uivrsi of Pakisa

254 Diffrial Equaios (MTH) W, W 5 ad W So ha u 5 ad u u d [ d] [ d] Thrfor ad u d. ( ) p H h gral soluio is: p Eampl 6 Solv d d l d d Soluio Cosidr h assoiad homogous diffrial quaio. d d d d or ( D D ) Wih h subsiuio, w hav: D, D ( ) So h homogous diffrial quaio boms: [ ( ) ] d d ( ) d d Puig, w g h auiliar quaio as: m m ( m ) m, l. m Now h o-homogous diffrial quaio boms: d d d d Coprigh Virual Uivrsi of Pakisa

255 Diffrial Equaios (MTH) B h mhod of udrmid offiis w r a pariular soluio of h form A B. This assumpio lads o p B A B so ha A ad B Usig p, w g ; So h gral soluio of h origial diffrial quaio o h irval (, ) is l l 7. Eriss Solv usig d d d d d d d d d d d d 5 d d 5 d d d d 6 d d d d 6 d d d d, (), '() d d d d, (), '() d d d d 8 d d d d 5 9 d d Coprigh Virual Uivrsi of Pakisa

256 Diffrial Equaios (MTH) 8 Powr Sris (A Iroduio) A sadard hiqu for solvig liar diffrial quaios wih variabl offiis is o fid a soluio as a ifii sris. Of his soluio a b foud i h form of a powr sris. Thrfor, i his lur w disuss som of h mor impora fas abou powr sris. Howvr, for a i-dph rviw of h ifii sris op o should osul a sadard alulus. 8. Powr Sris A powr sris i( a ) is a ifii sris of h form ( a) ( a) ( a). Th offiis,,, ad a ar osas ad rprss a variabl. I his disussio w will ol b ord wih h ass whr h offiis, ad a ar ral umbrs. Th umbr a is kow as h r of h powr sris. \ Eampl Th ifii sris ( ) is a powr sris i. This sris is rd a zro. 8. Covrg ad Divrg Eampl If w hoos a spifid valu of h variabl h h powr sris boms a ifii sris of osas. If, for h giv, h sum of rms of h powr sris quals a fii ral umbr, h h sris is said o b ovrg a. A powr sris ha is o ovrg is said o b a divrg sris. This mas ha h sum of rms of a divrg powr sris is o qual o a fii ral umbr. (a) Cosidr h powr sris!!!!!! Si for h sris bom Thrfor, h powr sris ovrgs o h umbr (b) Cosidr h powr sris Coprigh Virual Uivrsi of Pakisa 5

257 Diffrial Equaios (MTH)!( ) ( )!( )!( ) Th sris divrgs, p a. For isa, if w ak h h sris boms!( ) 8 Clarl h sum of all rms o righ had sid is o a fii umbr. Thrfor, h sris is divrg a. Similarl, w a s is divrg a all ohr valus of 8.. Th Raio Ts To drmi for whih valus of a powr sris is ovrg, o a of us h Raio Ts. Th Raio s sas ha if a a ( a ) a lim lim - a L is a powr sris ad Th: Th powr sris ovrgs absolul for hos valus of for whih L <. Th powr sris divrgs for hos valus of for whih L > or L. Th s is iolusiv for hos valus of for whih L. 8.. Irval of Covrg Th s of all ral valus of for whih a powr sris ( ) a ovrgs is kow as h irval of ovrg of h powr sris. 8.. Radius of Covrg Cosidr a powr sris ( a) Th al o of h followig hr possibiliis is ru: Th sris ovrgs ol a is r a. Th sris ovrgs for all valus of. Thr is a umbr R > suh ha h sris ovrgs absolul saisfig a < R ad divrgs for a > R. This mas ha h sris ovrgs for ( a Ra, R) ad divrgs ou sid his irval. Coprigh Virual Uivrsi of Pakisa 6

258 Diffrial Equaios (MTH) Th umbr R is alld h radius of ovrg of h powr sris. If firs possibili holds h R ad i as of d possibili w wri R. From h Raio s w a larl s ha h radius of ovrg is giv b providd h limi iss. R lim 8.. Covrg a a Edpoi If h radius of ovrg of a powr sris is R >, h h irval of ovrg of h sris is o of h followig [ ] ( a Ra, R), ( a Ra, R], [ a Ra, R), a Ra, R To drmi whih of hs irvals is h irval of ovrg, w mus odu spara ivsigaios for h umbrs a Rad a R. Eampl Cosidr h powr sris a Th lim a lim a or a lim lim lim a Thrfor, i follows from h Raio Ts ha h powr sris ovrgs absolul for hos valus of whih saisf < This mas ha h powr sris ovrgs if blogs o h irval (,) Th sris divrgs ousid his irval i.. wh > or <. Th ovrg of h powr sris a h umbrs ad mus b ivsigad sparal b subsiuig io h powr sris. a) Wh w subsiu, w obai () whih is a divrg p -sris, wih p. Coprigh Virual Uivrsi of Pakisa 7

259 Diffrial Equaios (MTH) b) Wh w subsiu, w obai ( ) ( ) whih ovrgs, b alraig sris s. H, h irval of ovrg of h powr sris is [,). This mas ha h sris is ovrg for hos vals of whih saisf < Eampl Fid h irval of ovrg of h powr sris a ( ) Soluio Th powr sris is rd a ad h radius of ovrg of h sris is ( ) R lim H, h sris ovrgs absolul for hos valus of whih saisf h iquali < < < 5 (a) A h lf dpoi w subsiu i h giv powr sris o obai h sris of osas: a ( ) This sris is ovrg b h alraig sris s. (b) A h righ dpoi w subsiu 5 i h giv sris ad obai h followig harmoi sris of osas Si a harmoi sris is alwas divrg, h abov powr sris is divrg. H, h sris h irval of ovrg of h giv powr sris is a half op ad half losd irval[, 5 ). 8. Absolu Covrg Wihi is irval of ovrg a powr sris ovrgs absolul. I ohr words, o h sris of absolu valus ( a) ovrgs for all valus i h irval of ovrg. Coprigh Virual Uivrsi of Pakisa 8

260 Diffrial Equaios (MTH) 8. Powr Sris Rprsaio of Fuios A powr sris ( a) drmis a fuio f whos domai is h irval of ovrg of h powr sris. Thus for all i h irval of ovrg, w wri f ( ) ( a) ( a) ( a) ( a) If a fuio is f is dfid i his wa, w sa ha ( a) rprsaio for f( ). W also sa ha f is rprsd b h powr sris is a powr sris 8.. Thorm Suppos ha a powr sris ( a) has a radius of ovrg > vr i h irval of ovrg a fuio f is dfid b R ad for Th f ( ) ( a) ( a) ( a) ( a) Th fuio f is oiuous, diffriabl, ad igrabl o h irval ( a R, a R). f ad f ( ) d a b foud from rm-b-rm diffriaio ad graio. Thrfor Morovr, ( ) ( ) ( ) ( ) ( ) f a a a ( ) ( ) ( a) ( a) f d C a C ( a) Th sris obaid b diffriaio ad igraio hav sam radius of ovrg. Howvr, h ovrg a h d pois a R ad a R of h irval Coprigh Virual Uivrsi of Pakisa 9

261 Diffrial Equaios (MTH) ma hag. This mas ha h irval of ovrg ma b diffr from h irval of ovrg of h origial sris. Eampl 5 Fid a fuio f ha is rprsd b h powr sris ( ) Soluio Th giv powr sris is a gomri sris whos ommo raio is r. Thrfor, if < h h sris ovrgs ad is sum is H w a wri a S r ( ) This las prssio is h powr sris rprsaio for h fuio f( ). 8.. Sris ha ar Idiall Zro If for all ral umbrs i h irval of ovrg, a powr sris is idiall zro i.. o ( ), > a R Th all h offiis i h powr sris ar zro. Thus w a wri,,,, 8.5 Aali a a Poi A fuio f is said o b aali a poi a if h fuio a b rprsd b powr sris i ( a ) wih a posiiv radius of ovrg. Th oio of aalii a a poi will b impora i fidig powr sris soluio of a diffrial quaio. Eampl 6 Si h fuios, os, ad l( ) a b rprsd b h powr sris!! os l( ) Coprigh Virual Uivrsi of Pakisa 5

262 Diffrial Equaios (MTH) Thrfor, hs fuios ar aali a h poi. 8.6 Arihmi of Powr Sris Powr sris a b ombid hrough h opraios of addiio, mulipliaio, ad divisio. Th produr for addiio, mulipliaio ad divisio of powr sris is similar o h wa i whih polomials ar addd, muliplid, ad dividd. Thus w add offiis of lik powrs of, us h disribuiv law ad oll lik rms, ad prform log divisio. Eampl 7 If boh of h followig powr sris ovrg for ( ), ( ) f g b Th f ( ) g( ) ( b ) < R f g b b b b b b ad ( ) ( ) ( ) ( ) Eampl 8 Fid h firs four rms of a powr sris i for h produ Soluio: From alulus h Malauri sris for ad 6 os. Muliplig h wo sris ad ollig h lik rms ilds os. os ar, rspivl, os. 6 ( ) 6 6 Th irval of ovrg of h powr sris for boh h fuios ad os is,. Cosqul h irval of ovrg of h powr sris for hir produ,. ( ) os is also ( ) Eampl 9 Fid h firs four rms of a powr sris i for h fuio s. Coprigh Virual Uivrsi of Pakisa 5

263 Diffrial Equaios (MTH) Soluio W kow ha Thrfor usig log divisio, w hav 6 s, os os H, h powr sris for h fuio f( ) s is s Th irval of ovrg of his sris is ( π /, π / ) No ha. Th produrs illusrad i ampls ad ar obviousl dious o do b had. Thrfor, problms of his sor a b do usig a ompur algbra ssm (CAS) suh as Mahmaia. Wh w p h ommad: Sris S[ ], {,, 8 } ad r, h Mahmaia immdial givs h rsul obaid i h abov ampl. For fidig powr sris soluios i is impora ha w bom adp a simplifig h sum of wo or mor powr sris, ah sris prssd i summaio (sigma) oaio, o a prssio wih a sigl. This of rquirs a shif of h summaio idis. Coprigh Virual Uivrsi of Pakisa 5

264 Diffrial Equaios (MTH) I ordr o add a wo powr sris, w mus sur ha: (a) Tha summaio idis i boh sris sar wih h sam umbr. (b) Tha h powrs of i ah of h powr sris b i phas. Thrfor, if o sris sars wih a mulipl of, sa, o h firs powr, h h ohr sris mus also sar wih h sam powr of h sam powr of. Eampl Wri h followig sum of wo sris as o powr sris 6 Soluio To wri h giv sum powr sris as o sris, w wri i as follows: 6 6 Th firs sris o righ had sid sars wih for ad h sod sris also sars wih for. Boh h sris o h righ sid sar wih. To g h sam summaio id w ar ispird b h pos of whih is i h firs sris ad i h sod sris. Thrfor, w l k, k i h firs sris ad sod sris, rspivl. So ha h righ sid boms: k k ( ) k k. k 6( k) k k Rall ha h summaio id is a dumm variabl. Th fa ha k i o as ad k i h ohr should aus o ofusio if ou kp i mid ha i is h valu of h summaio id ha is impora. I boh ass k aks o h sam sussiv valus,,, for,,, (for k )ad,,, (for k ) W ar ow i a posiio o add h wo sris i h giv sum rm b rm: k 6 ( k ) k 6( k ) k k If ou ar o ovid, h wri ou a fw rms o boh sris of h las quaio. Coprigh Virual Uivrsi of Pakisa 5

265 Diffrial Equaios (MTH) 9 Powr Sris Soluio of a Diffrial Equaio W kow ha h plii soluio of h liar firs-ordr diffrial quaio d d is Also 6 If w rpla b i h sris rprsaio of diffrial quaio as!, w a wri h soluio of h This las sris ovrgs for all ral valus of. I ohr words, kowig h soluio i adva, w wr abl o fid a ifii sris soluio of h diffrial quaio. W ow propos o obai a powr sris soluio of h diffrial quaio dirl; h mhod of aak is similar o h hiqu of udrmid offiis. Eampl d Fid a soluio of h DE: i h form of powr sris i. d Soluio If w assum ha a soluio of h giv quaio iss i h form Th qusio is ha: Ca w drmi offiis for whih h powr sris ovrgs o a fuio saisfig h diffrial quaio? Now rm-b-rm diffriaio of h proposd sris soluio givs d d Usig h las rsul ad h assumd soluio, w hav d d W would lik o add h wo sris i his quaio. To his d w wri Coprigh Virual Uivrsi of Pakisa 5

266 Diffrial Equaios (MTH) d d ampl b lig k, k Thrfor, las quaio boms ( ) Afr w add h sris rm wis, i follows ha d d [( k ) ] k k ad h prod as i h prvious i h firs ad sod sris, rspivl. k k k k k d k d k Subsiuig i h giv diffrial quaio, w obai k ( k ) k k k I ordr o hav his ru, i is ssar ha all h offiis mus b zro. This mas ha ( ) k k k, k, k,,, This quaio provids a rurr rlaio ha drmis h offii k. Si k for all h idiad valus of k, w a wri as k k k Iraio of his las formula h givs k, k, k,! k, 5 5 k 5, 6 6!! k 6, k Coprigh Virual Uivrsi of Pakisa 55

267 Diffrial Equaios (MTH) k 7, 8 6 8!! ad so o. Thus from h origial assumpio (7), w fid 5 5 6!! 6!!! Si h offii rmais ompll udrmid, w hav i fa foud h gral soluio of h diffrial quaio. No ha Th diffrial quaio i his ampl ad h diffrial quaio i h followig ampl a b asil solvd b h ohr mhods. Th poi of hs wo ampls is o prpar ourslvs for fidig h powr sris soluio of h diffrial quaios wih variabl offiis. Eampl Fid soluio of h d: i h form of a powrs sris i. Soluio W assum ha a soluio of h giv diffrial quaio iss i h form of Th rm b rm diffriaio of h proposd sris soluio ilds ( ) Subsiuig h prssio for ad, w obai ( ) Noi ha boh sris sar wih. If w, rspivl, subsiu k, k, k,,, Coprigh Virual Uivrsi of Pakisa 56

268 Diffrial Equaios (MTH) i h firs sris ad sod sris o h righ had sid of h las quaio. Th w afr usig, i ur, k ad k, w g k ( ) ( ) k k k k k k k k or ( )( ) k k Subsiuig i h giv diffrial quaio, w obai k From his las idi w olud ha or ( )( ) k k k k ( k )( k ) k k k,,,, ( k )( k ) k k From iraio of his rurr rlaio i follows ha...!..!...!.5..5!.6.5.6!.7.6.7! ad so forh. This iraio lavs boh ad arbirar. From h origial assumpio w hav !.!.!.5!.6!.7! k k k k Coprigh Virual Uivrsi of Pakisa 57

269 Diffrial Equaios (MTH) or !.! 6.6!.!.5! 6.7! is a gral soluio. Wh h sris ar wri i summaio oaio, ( ) k k ( ) k ( k )! ad ( ) k ( ) ( k ) k! h raio s a b applid o show ha boh sris ovrgs for all. You migh also rogiz h Malauri sris as ( ) os ( / ) ad ( ) ( / ) si. 9. Eris Fid h irval of ovrg of h giv powr sris..... k k k k k k ( ) k k k 7! k k k k Fid h firs four rms of a powr sris i for h giv fuio. 5. si 6. l( ) k Solv ah diffrial quaio i h mar of h prvious haprs ad h ompar h rsuls wih h soluios obaid b assumig a powr sris soluio Coprigh Virual Uivrsi of Pakisa 58

270 Diffrial Equaios (MTH) Soluio abou Ordiar Pois. Aali Fuio A fuio f is said o b aali a a poi a if i a b rprsd b a powr sris i (-a) wih a posiiv radius of ovrg. Suppos h liar sod-ordr diffrial quaio a ) a ( ) a ( ) () ( is pu io h form P( ) Q( ) () b dividig b h ladig offii a ( ).. Ordiar ad sigular pois A poi is said o b a ordiar poi of a diffrial quaio () if boh P() ad Q() ar aali a. A poi ha is o a ordiar poi is said o b sigular poi of h quaio... Polomial Coffiis If a ( ), a ( ) ad a ( ) ar polomials wih o ommo faors, h is Eampl (i) a ordiar poi if a ( ) or (ii) a sigular poi if a ( ). (a) Th sigular pois of h quaio ( ) 6 ar h soluios of or ±. All ohr fii valus of ar h ordiar pois. (b) Th sigular pois d o b ral umbrs. Th quaio ( ) 6 has h sigular pois a h soluios of, aml, ± i. All ohr fii valus, ral or ompl, ar ordiar pois. Eampl Th Cauh-Eulr quaio a b, whr a, b ad ar osas, has sigular poi a. All ohr fii valus of, ral or ompl, ar ordiar pois. Coprigh Virual Uivrsi of Pakisa 59

271 Diffrial Equaios (MTH). Thorm (Eis of Powr Sris Soluio) If is a ordiar poi of h diffrial quaio P( ) Q( ), w a alwas fid wo liarl idpd soluios i h form of powr sris rd a : ( ). A sris soluio ovrgs a las for < R, whr R is h disa from o h loss sigular poi (ral or ompl). Eampl Solv. Soluio W s ha is a ordiar poi of h quaio. Si hr ar o fii sigular pois, hr is wo soluios of h form ovrg for <. Prodig, w wri ( ) ( ) ( ) boh sris sar wih Lig k i h firs sris ad k i h sod, w hav k k ( k )( k ) k k k k k k [( k )( k ) k k ] ad k )( k ) k Th las prssio is sam as k, ( k )( k ) ( k k k,,, Coprigh Virual Uivrsi of Pakisa 6

272 Diffrial Equaios (MTH) Iraio givs baus , ad so o. I is obvious ha boh ad ar arbirar. Now [ ] [ ] Coprigh Virual Uivrsi of Pakisa 6

273 Diffrial Equaios (MTH) Eampl Solv Soluio ( ). Si h sigular pois ar ± i, ovrg a las for <. Th assumpio is h ordiar poi, a powr sris will lads o ( ) ( ) ( ) ( ) 6 ( ) ( ) k k- k k 6 [ ( ) ( )( ) ] k k k k k k k k k k k or 6 [( )( ) ( )( ) ] k k k k k k k. k Thus This implis ( k )( k ) k ( k )( k ) k k ( k ) k, ( k ) Iraio of h las formula givs k,,! Coprigh Virual Uivrsi of Pakisa 6

274 Diffrial Equaios (MTH) ! ! ad so o ! Thrfor [ ]!!! 55! Th soluios ar 5 ( ) ( ) [ ( ) ],! ( ). < Eampl If w sk a soluio for h quaio ( ), w obai ad h hr-rm rurr rlaio k k, ( k )( k ) k k,,, To simplif h iraio w a firs hoos, ; his ilds o soluio. Th ohr soluio follows from hoosig,. Wih h firs assumpio w fid Coprigh Virual Uivrsi of Pakisa 6

275 Diffrial Equaios (MTH) Thus o soluio is 6 ad so o [ ] 5 ( ) [ ]. 6 Similarl if w hoos, h H aohr soluio is 6 ad so o Eah sris ovrgs for all fii valus of.. No-polomial Coffiis 5 ( ) [ ]. 6 Th ampl illusras how o fid a powr sris soluio abou a ordiar poi of a diffrial quaio wh is offiis ar o polomials. I his ampl w s a appliaio of mulipliaio of wo powr sris ha w disussd arlir. Eampl Solv (os ) Soluio Th quaio has o sigular poi. 6 Si os, i is s ha is a ordiar poi.!! 6! Thus h assumpio lads o Coprigh Virual Uivrsi of Pakisa 6

276 Diffrial Equaios (MTH) (os ) ( ) ( )!! 6 ( 6 5 ) ( )( )!! 6! (6 ) ( ) ( 5 ) If h las li b idiall zro, w mus hav Now or ad so o ( ) ( ) 6 ( ) [ ] ad ad ar arbirar. 6 5 ( ) [ ] Si h diffrial quaio has o sigular poi, boh sris ovrg for all fii valus of..5 Eris I ah of h followig problms fid wo liarl idpd powr sris soluios abou h ordiar poi..... ( ) 5. ( ) 6 Coprigh Virual Uivrsi of Pakisa 65

277 Diffrial Equaios (MTH) Soluios abou Sigular Pois If is sigular poi, i is o alwas possibl o fid a soluio of h form ( ) for h quaio a( ) a( ) a( ) Howvr, w ma b abl o fid a soluio of h form ( ) r, whr r is osa o b drmid. To dfi rgular/irrgular sigular pois, w pu h giv quaio io h sadard form P ( ) Q ( ). Rgular ad Irrgular Sigular Pois A Sigular poi of h giv quaio a( ) a( ) a( ) is said o b a rgular sigular poi if boh ( ) P( ) ad ( ) Q ( ) ar aali a. A sigular poi ha is o rgular is said o b a irrgular sigular poi of h quaio... Polomial Coffiis If h offiis i h giv diffrial quaio a ( ) a ( ) a ( ) ar polomials wih o ommo faors, abov dfiiio is quival o h followig: a( ) a ( ) L a( ) Form P () ad Q() b rduig ad o lows a ( ) a ( ) rms, rspivl. If h faor ( ) appars a mos o h firs powrs i h domiaor of P () ad a mos o h sod powr i h domiaor of Q(), h is a rgular sigular poi. Eampl ± Dividig h quaio b ( )( ) ar sigular pois of h quaio ( ) ( ) ( ( ) ( ) P ( ) ad Q ( ) ) ( ) ( ), w fid ha. is a rgular sigular poi baus powr of i P () is ad i Q() is.. is a irrgular sigular poi baus powr of Th s odiio is violad. Eampl Boh ad ar sigular pois of h diffrial quaio. i P () is. Coprigh Virual Uivrsi of Pakisa 66

278 Diffrial Equaios (MTH) ( ) ( ) Baus ( ) or,- Now wri h quaio i h form ( ) ( ) or So ( ) ( ) P ( ) ad Q ( ) ( ) ( ) Shows ha is a irrgular sigular poi si ( ) appars o h sod powrs i h domiaor of P (). No, howvr, is a rgular sigular poi. Eampl a) ad ar sigular pois of h diffrial quaio ( ) Baus or ±. Now wri h quaio i h form ( ) or ( )( ) ( )( ) P ( ) ( )( ) ad Q ( ) ( )( ) Clarl ± ar rgular sigular pois. (b) is a irrgular sigular pois of h diffrial quaio or givig Q ( ). () is a rgular sigular pois of h diffrial quaio Coprigh Virual Uivrsi of Pakisa 67

279 Diffrial Equaios (MTH) 5 Baus h quaio a b wri as 5 Q( ). 5 givig P( ) ad I par () of Eampl w oid ha ( ) ad ( ) do o v appar i h domiaors of P () ad Q() rspivl. Rmmbr, hs faors a appar a mos i his fashio. For a sigular poi, a ogaiv powr of ( ) lss ha o (aml, zro) ad ogaiv powr lss ha wo (aml, zro ad o) i h domiaors of P ( ) ad Q (), rspivl, impl is a rgular sigular poi. Plas o ha h sigular pois a also b ompl umbrs. For ampl, ± i ar rgular sigular pois of h quaio ( 9) ( ) Baus h quaio a b wri as 9 9 P( ). Q( ). ( i)( i) ( i )( i). Mhod of Frobius. To solv a diffrial quaio a( ) a( ) a( ) abou a rgular sigular poi w mplo h Frobius Thorm... Frobius Thorm If is a rgular sigular poi of quaio a( ) a( ) a( ), h hr iss a las o sris soluio of h form r r ( ) ( ) ( ) whr h umbr r is a osa ha mus b drmid. Th sris will ovrg a las o som irval < < R. No ha h soluios of h form Mhod of Frobius. Idif rgular sigular poi, ( ) r ar o guarad. Coprigh Virual Uivrsi of Pakisa 68

280 Diffrial Equaios (MTH). Subsiu ( ) r i h giv diffrial quaio,. Drmi h ukow po r ad h offiis.. For simplii assum ha. Eampl As is rgular sigular pois of h diffrial quaio. W r a soluio of h form Thrfor Ad ( r) whih implis r ( r ) r. r ( r)( r ) (. r. r r r )( r ) r ( r)( r) r [ r(r ) ( k r )(k r ) k, k,,,... k ( r ) - r. r ( r)( r ) ]. k k r k r(r ) [( k r )(k r ) k k] k Si ohig is gaid b akig, w mus h hav r ( r ) [alld h idiial quaio ad is roos r, ar alld idiial roos or pos of h sigulari.]. Coprigh Virual Uivrsi of Pakisa 69

281 Diffrial Equaios (MTH) k ad k, k,,,... ( k r )(k r ) Subsiu r ad r i h abov quaio ad hs valus will giv wo diffr rurr rlaios: For r, k k (k 5)( k ), k,,,... () For r k k ( k )(k ), k,,,... () Iraio of () givs 5. 8.!5.8.!5.8..! I gral,,,...!5.8...( ) Iraio of () givs..!..7!..7.!..7. I gral,,,...!..7...( ) Thus w obai wo sris soluios Coprigh Virual Uivrsi of Pakisa 7

282 Diffrial Equaios (MTH)!5.8...( ) (). ()!..7...( ) B h raio s i a b dmosrad ha boh () ad () ovrg for all fii valus of. Also i should b lar from h form of () ad () ha ihr sris is a osa mulipl of h ohr ad hrfor, ( ) ad ( ) ar liarl idpd o h - ais. H b h suprposiio priipl C C ( ) C ( )!5.8...( ) C!..7...( ), < is a ohr soluio of h diffrial quaio. O a irval o oaiig h origi, his ombiaio rprss h gral soluio of h diffrial quaio No: Th mhod of Frobius ma o alwas provid soluios. Eampl: Th diffrial quaio has rgular sigular poi a W r a soluio of h form r Thrfor ( r ) so ha r ad ( r )( r ) r. r k r( r ) [( k r )( k r ) k k] k so ha h idiial quaio ad po ar r ( r ) ad r, r, rspivl. Si ( k r )( k r ) k, k,,,... () k i follows ha wh r, k ( k )( k ) k, Coprigh Virual Uivrsi of Pakisa 7

283 Diffrial Equaios (MTH). Thus o sris soluio is!( )! Now wh r, () boms.!!.5!5!.6!6! [ ] ( k )( k ) k () k,,,...!( )!, <.!( )! bu o hr ha w do o divid b ( k )( k ) immdial si his rm is zro for k. Howvr, w us h rurr rlaio () for h ass k ad k :. ad. Th lar quaio implis ha ad so h formr quaio implis ha. Coiuig, w fid k k k,,... ( k )( k )..!.! 5,....5!.5! Coprigh Virual Uivrsi of Pakisa 7

284 Diffrial Equaios (MTH) I gral,,,5,... ( )!! Thus. () ( )!! Howvr, los ispio of () rvals ha is simpl osa mulipl of. To s his, l k i (). W olud ha h mhod of Frobius givs ol o sris soluio of h giv diffrial quaio.. Cass of Idiial Roos Wh usig h mhod of Frobius, w usuall disiguish hr ass orrspodig o h aur of h idiial roos. For h sak of disussio l us suppos ha r ad r ar h ral soluios of h idiial quaio ad ha, wh appropria, r dos h largs roo... Cas I (Roos o Diffrig b a Igr) If r ad r ar disi ad do o diffr b a igr, h hir is wo liarl idpd soluios of h diffrial quaio of h form r.., ad r b, b. Eampl 6 Solv ( ). Soluio: If ( ) r, h r ( r)( r ) r ( r) r ( r) r r ( r r)( r ) [ r(r ) ( r ) ( r)( r ) r ( r ) k k r k r(r ) [( k r )(k r ) k ( k r ) k] k whih implis r ( r ) ( k r )(k r ) ( k r ), k,,,... () k k ] Coprigh Virual Uivrsi of Pakisa 7

285 Diffrial Equaios (MTH) For r, w a divid b k i h abov quaio o obai I gral k ( k ) k,...!..! ( ),,,,...! ( )! Thus w hav [ ], whih ovrgs for.as giv, h sris is o maigful for < baus of h prs of. Now for r, () boms I gral Thus sod soluio is k k k ( ),,,, ( ) Coprigh Virual Uivrsi of Pakisa 7

286 Diffrial Equaios (MTH) ( ) ( ). <. O h irval (, ), h gral soluio is C ( ) C ( ). Soluios abou Sigular Pois. Mhod of Frobius (Cass II ad III) Wh h roos of h idiial quaio diffr b a posiiv igr, w ma or ma o b abl o fid wo soluios of havig form a ) a ( ) a ( ) () ( ) ( r If o, h o soluio orrspodig o h smallr roo oais a logarihmi rm. Wh h pos ar qual, a sod soluio alwas oais a logarihm. This lar siuaio is similar o h soluio of h Cauh-Eulr diffrial quaio wh h roos of h auiliar quaio ar qual. W hav h wo ass... Cas II (Roos Diffrig b a Posiiv Igr) If r r N, whr N is a posiiv igr, h hr is wo liarl idpd soluios of h form r, ( a ) r C( )l b, b ( b ) Whr C is a osa ha ould b zro. Equal Idiial Roos: If r r, hr alwas is wo liarl idpd soluios of () of h form r, ( a ) () Coprigh Virual Uivrsi of Pakisa 75

287 Diffrial Equaios (MTH) r ( )l b r r ( b ) Coprigh Virual Uivrsi of Pakisa 76

288 Diffrial Equaios (MTH) Eampl 7: Solv ( 6) () Soluio: Th assumpio ( 6) r lads o ( r)( r ) r 6 ( r) r ( r) r r r r( r 7) ( r)( r 7) ( r) r k r( r 7) [( k r )( k r 6) k ( k r ) k] k Thus r(r - 7) so ha r, r, r r 7, ad 7 ( k r )( k r 6) k ( k r ) k, k,,,,... () For smallr roo r,() boms k ( k )( k 6) ( k ) () ( k ) ( k )( k6) k k rurr rlaio boms k Si k-6, wh, k6, w do o divid b his rm uil k>6.w fid This implis ha.( 6) ( ).( 5) ( ).( ) ( )..(-). 5. (-) (-) , Bu ad 7 a b hos arbiraril. H 5 Coprigh Virual Uivrsi of Pakisa 77

289 Diffrial Equaios (MTH) () ad for k 7 ( k ) k k ( k )( k6) ! !8.9. () 56 ( ) 7, ( 7)!8 9 ( ) 8,9,, (5) If w hoos 7 ad I follows ha w obai h polomial soluio [ ], Bu wh 7 ad, I follows ha a sod, hough ifii sris soluio is 7 () 56 ( ) 7[ ] 8 ( 7)! 8 9 k 7 k 7[ ] k k!8 9 ( k 7) ()56 ( k ), < (6) Fiall h gral soluio of quaio () o h irval (, ) is Y ) ( ) ( k 7 ()56 ( k ) k 7 [ ] [ ] k! 8 9 ( k 7) I is irsig o obsrv ha i ampl 9 h largr roo r 7 wr o usd. Had w do so, w would hav obaid a sris soluio of h form* 7 (7) Coprigh Virual Uivrsi of Pakisa 78

290 Diffrial Equaios (MTH) Whr ar giv b quaio () wih r 7 ( k ) k k, k,,... ( k 8)( k ) Iraio of his lar rurr rlaio h would ild ol o soluio, aml h soluio giv b (6) wih plaig h rol of 7 ) Wh h roos of idiial quaio diffr b a posiiv igr, h sod soluio ma oai a logarihm. O h ohr had if w fail o fid sod sris p soluio, w a alwas us h fa ha p( ) d ( ) d (8) ( ) is a soluio of h quaio P( ) Q( ),whvr is a kow soluio. No: I as i is alwas a good ida o work wih smallr roos firs. Eampl8 Fid h gral soluio of Soluio Th mhod of Frobius provid ol o soluio o his quaio, aml,!( )! 6 From (8) w obai a sod soluio p( ) d ( ) d ( ) ( ) ( ) (9) d [ ] 6 d 7 ( ) [ ] 6 9 [ ] d 7 9 ( ) l ( )l ( )... 7 (*) 9 ( ) ( )l ( )... 7 (**) Coprigh Virual Uivrsi of Pakisa 79

291 Diffrial Equaios (MTH) Eampl 9 Fid h gral soluio of Soluio : l b () ()!( )! diffria() givs l ( )b l ( )( )b so ha l ( )( )b ( )b b ( )b b () whr w hav ombid hs wo summaios ad usd h fa ha Diffria() wa wri()as ( ) b!( )!!( )! b ( )!( )! ( ) b ( b b ) ( ) b b ( k ) k ( b b) kk ( ) bk bk. k k!( k )! () Sig () qual o zro h givs b b ad ( k ) k( k ) b k b k!( k )! k, For k,,, () Coprigh Virual Uivrsi of Pakisa 8

292 Diffrial Equaios (MTH) Wh k i quaio () w hav b b so ha bu b, b, bu b is arbirar Rwriig quaio () as bk ( k ) bk (5) k( k ) k!( k )! k( k ) ad valuaig for k,, givs b b b 9 b b ad so o. Thus w a fiall wri l b b b b b l b (6) 9 Whr b is arbirar. Equival Soluio: A his poi ou ma b wodrig whhr (*) ad (6) ar rall quival. If w hoos i quaio (**), h 8 8 l 5 l (7) l... 9 Whih is prisl obaid wha w obaid from (6). If b is hos as 6 Th ampl illusras h as wh h idiial roos ar qual. Coprigh Virual Uivrsi of Pakisa 8

293 Diffrial Equaios (MTH) Eampl : Fid h gral soluio of (8) r Soluio : Th assumpio lads o r r r (r)(r-) (r) Thrfor r r (r) r r (r) r k r (kr) k k k r, ad so h idiial roos ar qual: r r. Morovr w hav k r ) k, k,,, (9) ( k Clarl h roos r will ild o soluio orrspodig o h offiis dfid b h iraio of k k k,,, ( k ) Th rsul is, < () (!) ( ) ) d d ( ) ( ) d () 8 d 7 9 () 8... d Coprigh Virual Uivrsi of Pakisa 8

294 Diffrial Equaios (MTH) 7 () l 8 7 Thus o h irval (, ) h gral soluio of (8) is () ()l () 8... whr ( ) is dfid b () 7 7 I as II w a also drmi ( ) of ampl9 dirl from assumpio (b) Eriss I problm - drmi h sigular pois of ah diffrial quaio. Classif ah h sigular poi as rgular or irrgular. ( ) ( 9) ( ) ( ) 5 ( ) 6 ) 6 ( 5) ( ) 7 ( 6) ( ) ( ) 8 ( ) 9 ( 5)( ) ( ) 7( 5) ( ) ( ) ( ) I problm - show ha h idiial roos do o diffr b a igr. Us h mhod of Frobius o obai wo liarl idpd sris soluios abou h rgular sigular poi Form h gral soluio o (, ) ( ) Coprigh Virual Uivrsi of Pakisa 8

295 Diffrial Equaios (MTH) 5. ( ) ( ) ( ). ( ). ( ) I problm - show ha h idiial roos diffr b a igr. Us h mhod of Frobius o obai wo liarl idpd sris soluios abou h rgular sigular poi Form h gral soluio o (, ).. 5. ( ) ( ) ( ).. ( ). Coprigh Virual Uivrsi of Pakisa 8

296 Diffrial Equaios (MTH) Bssl s Diffrial Equaio d d A sod ordr liar diffrial quaio of h form ( v ) is alld Bssl s diffrial quaio. Soluio of his quaio is usuall dod b ( ) J v d d ad is kow as Bssl s fuio. This quaio ours frqul i advad sudis i applid mahmais, phsis ad girig.. Sris Soluio of Bssl s Diffrial Equaio Bssl s diffrial quaio is ( ) If w assum ha So ha C v () r C C ( r) ( r)( r ) r r r ( v ) C ( r)( r ) C ( r) C r v ( ) ( )( ) ( ) r r r Co r v C r r r v C C r r () From () w s ha h idiial quaio is r v, so h idiial roos ar r v r v. Wh r v h () boms, v v ( ) C v C v C C v C k k v ( ) ( ) Coprigh Virual Uivrsi of Pakisa 85

297 Diffrial Equaios (MTH) W a wri Th hoi C i () implis so for Thus v k ( vc ) ( k )( k vc ) k Ck k ( vc ) ( k )( k v) C k C k Ck C () k k,,, C C C ( k )( k v) 5 k,,, w fid, afr lig k,,,, ha C C ( v) C C ( v) C C C ( v) ( v)( v) C C C 6 6 ( v) ( v)( v)( v) C! v v v,,, ( ) C ( )( ) ( ) I is sadard prai o hoos C o b a spifi valu aml C v Γ ( v) whr Γ ( v) h Gamma fuio. Also Γ ( α) αγ ( α). () (5) Coprigh Virual Uivrsi of Pakisa 86

298 Diffrial Equaios (MTH) Usig his propr, w a rdu h idiad produ i h domiaor of (5) o o rm. For ampl H w a wri (5) as So h soluio is If C v ( ) ( ) Γ ( v ) v Γ v ( ) ( )( ) Γ ( v ) v Γ ( v) ( ) v v Γ ( v) ( ) ( )( ) ( )! v v v Γ ( v),,,, v! Γ ( v ) v v ( ) C! Γ ( v ) v, h sris ovrgs a las o h irval [ ). Bssl s Fuio of h Firs Kid As for r v, w hav Also for h sod po Jv ( ). v ( ) (6) (!) Γ ( v ) r v, w hav Jv Th fuio J ( ) ad ( ) ( ) v ( ) (7) (!) Γ( v ) v J v ar alld Bssl fuio of h firs kid of ordr v ad v rspivl. Now som ar mus b ak i wriig h gral soluio of (). Wh v, i is lar ha (6) ad (7) ar h sam. If v > ad r r v ( v) v is o a posiiv igr, h J v ( ) ad J v ( ) ar liarl idpd soluios of () o (, ) ad so h gral soluio of h irval would b If ( ) C ( ) C J Jv v r r v is a posiiv igr, a sod sris soluio of () ma iss. Eampl Fid h gral soluio of h quaio Coprigh Virual Uivrsi of Pakisa 87

299 Diffrial Equaios (MTH) o (, ) Soluio Th Bssl diffrial quaio is Comparig () ad (), w g ( v ) () () v, hrfor v ± So gral soluio of () is C J ( ) C ( ) / J / Eampl Fid h gral soluio of h quaio: 9 Soluio: W idif v, hrfor 9 v ± So gral soluio is C J ( ) C ( ) / J / Eampl Driv h formula J ( ) vj ( ) J ( ) Soluio As J ( ) v v ( )! Γ ( v ) v v v v ( ) ( v) J v ( )! Γ ( v ) v v ( ) ( ) v! Γ ( v )! ( v ) Γ v ( ) vjv ( ) ( )! Γ ( v ) k k vjv ( ) k! Γ ( v k) k ( ) J ( ) vj v v k v ( ) Coprigh Virual Uivrsi of Pakisa 88

300 Diffrial Equaios (MTH) So ( ) ( ) ( ) J J J v v v v Eampl Driv h rurr rlaio ( ) ( ) ( ) J J J Soluio: As ( ) ( ) ( ) s s s! s s! J ( ) ( ) ( ) ( )!! s s s J s s s ( ) ( ) ( )!! s s s s s s s ( ) ( ) ( ) ( ) ( ) s s s s s s s! s s! s! s s! ( ) ( )( ) ( ) ( ) ( ) ( ) s s s s s s! s! s s s s! s s s! ( ) ( ) ( ) ( ) ( ) s s s s s s! s! s! s s! ( ) ( ) ( ) ( ) s s s! s! s J Pu p s i d rm p s ( ) ( ) ( ) ( ) p p p! p p! J ( ) ( ) ( ) p p p p! p! J ( ) ( ) ( ) J J J Coprigh Virual Uivrsi of Pakisa 89

301 Diffrial Equaios (MTH) J ( ) J ( ) J ( ) Eampl 5 Driv h prssio of J ( ) for Soluio: Cosidr J ( ) As! Γ ( ) s s! s ( ) ( s)! s s ( ) J ( ) Γ ( s ) Γ ( s ) s pu / J / ( ) s ( ) Γ ( s ) Γ (/ s ) s Epadig R.H.S of abov J/ ( ) ( ) Γ ( s ) Γ ( s / ) s s ± s s s ( ) ( ) ( ) Γ ( ) Γ ( / ) Γ ( ) Γ ( / ) ( ) ( ) ( ) ( ) Γ ( ) Γ ( / ) Γ ( ) Γ ( / ) π π 5 π 5/ 9/ π 5 5/ 9/ π / 9/ π 5 Coprigh Virual Uivrsi of Pakisa 9

302 Diffrial Equaios (MTH) π 6 5 π! 5! si π J / ( ) si π Similarl for /, w prod furhr as bfor, ( ) s s! s ( ) ( s)! s J whr! Γ ( ) s s ( ) J ( ) Γ ( s ) Γ ( s ) s pu J J / / ( ) ( ) s ( ) Γ ( s ) Γ ( / s ) s ( ) Γ ( s ) Γ ( s /) s s s s Epadig h R.H.S of abov w g J/ ( ) Γ ( ) Γ ( / ) Γ ( ) Γ ( / ) ( ) ( ) Γ ( ) Γ ( / ) ( ) ( ) ( ) Coprigh Virual Uivrsi of Pakisa 9

303 Diffrial Equaios (MTH) J / ( ) 7 Γ() Γ(/ ) Γ() Γ(/ ) Γ() Γ(5/ ) Rmarks: J 7 () (/ ) Γ Γ(/ ) Γ(/ ) / 7/ Γ(/ ) / 7/ / 7 / / 7 / π / 7 / π 6 / 7 / π 8 / 7 / π / 7 / π π!! π os os!! / π ( ) os Bssl fuios of id half a odd igr ar alld Sphrial Bssl fuios. Lik ohr Bssl fuios sphrial Bssl fuios ar usd i ma phsial problms. Coprigh Virual Uivrsi of Pakisa 9

304 Diffrial Equaios (MTH) Eris Fid h gral soluio of h giv diffrial quaio o (, ).. 9. ( ). ( ). ( ) Eprss h giv Bssl fuio i rms of si ad os, ad powr of. 5. J ( ) / 6. J ( ) 5/ 7. J ( ) 7/ Coprigh Virual Uivrsi of Pakisa 9

305 Diffrial Equaios (MTH) Lgdr s Diffrial Equaio A sod ordr liar diffrial quaio of h form ( ) ( ) is alld Lgdr s diffrial quaio ad a of is soluio is alld Lgdr s fuio. If is posiiv igr h h soluio of Lgdr s diffrial quaio is P. alld a Lgdr s polomial of dgr ad is dod b ( ) W assum a soluio of h form C k ( ) ( ) k ( ) ( ) ( ) k C k k k k k k k k k Ckk Ck C k k k k k k k ( k ) Ck k( k ) Ck k ( ) Ck k [ ( ) C C ] [ ( ) C C 6C ] Ckk( k ) k k ( k ) Ckk ( ) Ckk k j k k j k ( ) ( )( ) k Ck k j k C C C 6C j ( j )( j ) Cj ( j)( j ) Cj j C C ( ) or k ( )( ) C C 6 ( )( ) ( )( ) C j k k j k k j j C j j C, j,,,... ( ) C! ( )( ) C! C j Coprigh Virual Uivrsi of Pakisa 9

306 Diffrial Equaios (MTH) From Iraio formula () C C 5 ( j)( j ) ( j )( j ) Cj Cj; j,, ( )( ) ( )( )( )( ) C ( )( ) ( )( )( )( ) 5 C! 5! C C () ( )( 5) ( )( ) ( )( )( 5) C C C 5 6 6! C 6 7 ( 5)( 6) ( 5)( )( )( )( )( 6) 7 6 C 5 ad so o. Thus a las <, w obai wo liarl idpd powr sris soluios. ( ) ( ) C C 7! ( ) ( ) ( )( )!! ( )( ) ( )( )( 5) 6! 6 ( )( ) ( )( )( )( )! 5! ( 5)( )( )( )( )( 6) No ha if is v igr, h firs sris rmias, whr ( ) For ampl if, h 7! 7 5 C is a ifii sris. ( ) C C 5! 5 7! rmias wih.i. wh is a o-gaiv igr, w obai a h-dgr polomial soluio of Lgdr s quaio. Si w kow ha a osa mulipl of a soluio of Lgdr s quaio is also a soluio, i is radiioal o hoos spifi valus for C ad C dpdig o whhr is v or odd posiiv igr, rspivl. For, w hoos C ad for,,6, Similarl, wh is a odd igr, h sris for ( ) 5 Coprigh Virual Uivrsi of Pakisa 95

307 Diffrial Equaios (MTH) C ( ) / ( ) ( ) Whras for, w hoos C ad for,5,7, C ( ) ( ) ( ) For ampl, wh, w hav 5 ( ) ( ) / ( ) ( 5 ). Lgdr s Polomials 8 P. ad from h abov hois of C ad C, w fid ha h firs svral Lgdr s polomials ar Lgdr s Polomials ar spifi h dgr polomials ad ar dod b ( ) From h sris for ( ) ad ( ) P quaios P P ( ) P ( ) No ha ( ) ( ), ( ), ( ), P P ( ) ( ) ( ) ( 5 ) P P, P ( ) ( 5 ) 8 ( ) 5 ( 6 7 ) P5 5 8 ( ) ( ) ar, i ur pariular soluio of h diffrial Coprigh Virual Uivrsi of Pakisa 96

308 Diffrial Equaios (MTH) ( ) 6 ( ). Rodrigus Formula for Lgdr s Polomials Th Lgdr Polomials ar also grad b Rodrigus formula d P! d ( ) ( ). Graig Fuio For Lgdr s Polomials Th Lgdr s polomials ar h offii of ( ) z z φ i asdig powrs of z. Now ( ) φ z z z( z) Thrfor b Biomial Sris φ { } z i h pasio of 5 z( z) { z( z) } { z( z) }!! 5 z z z z z z 8 z z 6z 8 6 ( ) ( ) ( ) z z z z z z z z z ( ) ( ) ( ) z z 5 z 5 z () 8 Also ( ) ( ) ( ) ( ) ( ) P z P P z P z P z Equaig Coffiis of () ad () Coprigh Virual Uivrsi of Pakisa 97

309 Diffrial Equaios (MTH) ( ) ( ) P P Whih ar Lgdr s Polomials. Rurr Rlaio P ( ) ( ) P ( ) ( 5 ) P 5 8 ( ) ( ) Rurr rlaios ha rla Lgdr s polomials of diffr dgrs ar also vr impora i som asps of hir appliaio. W shall driv o suh rlaio usig h formula ( ) ( ) P Diffriaig boh sids of () wih rsp o givs so ha afr muliplig b () ( ) ( ) ( ) ( ) P P, w hav ( )( ) ( ) ( ) P ( ) ( ) ( ) ( ) P P P( ) P( ) P( ) P( ) P ( ) P( ) P( ) P( ) P( ) P( ) Coprigh Virual Uivrsi of Pakisa 98

310 Diffrial Equaios (MTH) Obsrvig h appropria allaios, simplifig ad hagig h summaio idis k Equaig h oal offii of ( ) ( ) ( ) ( ) ( ) k k Pk k Pk kpk ( ) ( ) ( ) ( ) ( ) k o zro givs h hr-rm rurr rlaio k P k P kp, k,,, k k k.5 Orhogoall of Lgdr s Polomials Proof: Lgdr s Diffrial Equaio is ( ) ( ) L P ( ) ad Pm ( ) whih w a wri ar wo soluios of Lgdr s diffrial quaio h ( ) P ( ) P ( ) ( ) P ( ) ( ) Pm ( ) Pm ( ) m( m ) Pm ( ), ad ( ) P ( ) ( ) P ( ) () ( ) Pm ( ) m( m ) Pm ( ) Muliplig () b Pm ( ) ad () b P ( ) Now ad subraig, w g ( ){( ) } ( ){( ) ( )} Pm P P Pm { ( ) m( m )} P ( ) P ( ) m ' ' Add ad subra( ) P mp o formulizhabov ' ' ' ' { } { } Pm( ) ( ) P P( ) ( ) Pm () () Coprigh Virual Uivrsi of Pakisa 99

311 Diffrial Equaios (MTH) ( ) Pm ( ) P ( ) Pm ( ) ( ) P ( ) ( ) Pm ( ) P ( ) P ( ) ( ) Pm ( ) ( )[ ( ) ( ) m ( ) ( ) ] P P P P Whih shows ha () a b wri as ( ) Pm ( ) P ( ) Pm ( ) P ( ) { } ( ) ( ) ( ) ( ) m m Pm P ( ( ){ Pm( ) P ( ) Pm ( ) P} ) ( m)( m ) Pm( ) P( ) ( m) ( m ) P m ( ) P ( ) ( ( ){ Pm ( ) P ( ) Pm ( ) P ( ) }) b b ( ) ( ) m ( ) ( ) ( ( ){ m ( ) ( ) m ( ) ( )}) m m P P d P P P P d a a m b m Pm P d Pm P Pm P a { } ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) As for ± so m m Pm P d for ± ( ) ( ) ( ) ( ) Si m & ar o-gaiv Pm ( ) P ( ) d for m whih shows ha Lgdr s Polomials ar orhogoal w.r.o h wigh fuio w( ) ovr h irval [ ] b a Coprigh Virual Uivrsi of Pakisa

312 Diffrial Equaios (MTH).6 Normali odiio for Lgdr Polomials Cosidr h graig fuio Also Muliplig () ad () Igraig from - o P m () m ( ) m ( ) P () ( ) ( ) ( ) m ( ) ( ) m P P m ( ) m d Pm ( ) P ( ) d m m d P ( ) ( ) m P d ( ) m l ( ) m P m ( ) P ( ) d m Pm P d m l ( ) l ( ) m ( ) ( ) l ( ) l ( ) { ( ) ( )} l l Coprigh Virual Uivrsi of Pakisa

313 Diffrial Equaios (MTH) l ( ) l ( ) m Pm ( ) P ( ) d 5 m for m P ( ) P ( ) d 5 ( ) ( ) P ( ) d ( ) ( ) ( ) Equaig offii of o boh sids P ( ) d P ( ) P ( ) d P ( ) P ( ) d Coprigh Virual Uivrsi of Pakisa

314 Diffrial Equaios (MTH) whih shows ha Lgdr polomials ar ormal wih rsp o h wigh fuio w( ) ovr h irval < <. Rmark: Orhogali odiio for P ( ).7 Eris a also b wri as P ( ) P ( ) d δm, whr δm,, if m,ohrwis. Show ha h Lgdr s quaio has a alraiv form d d ( ) ( ) d d. Show ha h quaio si θ d d os ( ) ( si ) θ d dθ θ a b θ rasformd io Lgdr s quaio b mas of h subsiuio osθ. Us h plii Lgdr s polomials P ( ), P ( ), P ( ), ad P ( ) o valua P d for,,,. Graliz h rsuls.. Us h plii Lgdr polomials P ( ), P ( ), P ( ), ad P ( ) P P m d for m. Graliz h rsuls. o valua ( ) ( ) 5. Th Lgdr s polomials ar also grad b Rodrigus formula d P ( ) ( )! d vrif h rsuls for,,,. Coprigh Virual Uivrsi of Pakisa

315 Diffrial Equaios (MTH) 5 Ssms of Liar Diffrial Equaios Rall ha h mahmaial modl for h moio of a mass aahd o a sprig or for h rspos of a sris lrial irui is a diffrial quaio. d d a b f ( ) d d Howvr, w a aah wo or mor sprigs oghr o hold wo masss m ad m. Similarl a work of paralll iruis a b formd. To modl hs lar siuaios, w would d wo or mor oupld or simulaous quaios o dsrib h moio of h masss or h rspos of h work. Thrfor, i his lur w will disuss h hor ad soluio of h ssms of simulaous liar diffrial quaios wih osa offiis. No ha A hordr liar diffrial quaio wih osa offiis a, a,, a is a quaio of h form d d d a a a a g ( ) d d d d d d If w wri D, D,, D h his quaio a b wri as follows d d d ( ) ( ad a ) ( ) D ad a g 5. Simulaous Diffrial Equaios Th simulaous ordiar diffrial quaios ivolv wo or mor quaios ha oai drivaivs of wo or mor ukow fuios of a sigl idpd variabl. Coprigh Virual Uivrsi of Pakisa