MAT244H1a.doc. A differential equation is an equation involving some hypothetical function and its derivatives.

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1 MATHdo Iodio INTRODUCTION TO DIFFERENTIAL EQUATIONS A diffil qio is qio ivolvig som hpohil fio d is divivs Empl is diffil qio As sh, h diffil qio is dsipio of som fio (iss o o A solio o diffil qio is fio h sisfis h diffil qio Empl is solio o Som diffil qios fmos/impo:,, os, os, Rll h diffil qio dsibs phomo i ms of hgs Fo mpl, if dp F Fd dm v o d P mv, h Empl A pool ois V lis of w whih ois M g of sl P w s h pool os of v lis p mi, d f miig, is h sm Wi diffil qio h dsibs h dsi of sl i h pool bi im ρ L M V d ρ b h dsi im Th ρ ρ ρ ρ ρ To modl hg i, l ρ ρ ρ( ρ v V V v Th ( ρ ρ V ρv V ρ V ρ o dρ v v ρ ρ ρ Now, s, d V o V Wiho solvig his qio, w pdi fs bo his ssm As, dρ v d To solv his diffil qio, wi ρ V Igig boh sids, w g l ρ v V C ρ v C V C v V A v V ρ, so Pg of

2 MATHdo ρ ρ If w dd o ρ ρ v V, h w hv IVP (iiil vl poblm ISSUES ABOUT THE USE OF DIFFERENTIAL EQUATIONS How o sl l poblm o diffil qio Kp o s op! Som ps of ill-dfid Us diff pois of viw d diff ml diffil qio modls o foml hm Diffil qios hv ifiil m solios Whih o is os? Th iiil vl ml impo Th m b o li solio fod Is h solio? Is his solio iq? If mi sw is qid, i h vl of h solio o pil poi, h s mil ppoimios I dos o giv fligs fo h p, o dos i giv lbow oom Us hoil lsis if o d h bhvio of h solio This dos o giv vls To ow h bhvio loll/i ighbohood, solv i sis Th d dos o fi o solio Yo d o p (s i fdb/oolld ssm NOTATIONS WITH REGARD TO THE INPUT/OUTPUT SYSTEMS Empl si b wi s L[ ] L [ ] is h bl bo ssm is h ip Solv i, d h sw is h op Fo hoil pposs, mhmiis s hs qivls: (, o (,,,, f, F si is h sm s LINEAR VS NON-LINEAR DIFFERENTIAL EQUATIONS is li is o-li Fis Od Diffil Eqios LINEAR EQUATIONS Pg of

3 MATHdo Fis od li qios hv h fom p g Divio Sppos I fid µ so h µ ( p µ Mlipl boh sids of p g b µ : µ ( µ p µ g µ ( µ µ g, i ( µ ( µ g whih is Ig boh sids: µ ( µ g d C, so ( g d C µ µ µ B wh is µ? Si µ p µ p, hfo l µ µ p d So p ( d µ Gl Solio p g, To solv L p d µ (o os dd Th solio is [ g d C] µ µ Empl Solv H, p, g L d µ Th solio is [ d C] [ d C] ( C No h s, C Impo of Alsis O ds o hv dsd of h solio bfo (v f solvig i, wih sp o: Bhvio of h solio s Th d bhvio of h solio wihi fmil (dpds o Viio of Pm Rll h h fmil of solios of fis od li diffil qio p g, p ( C [ µ g d C] µ g d µ µ Noi h h fmil of solios is gd b µ µ This lds o h hiq of viio of pm Rll diffil qio L [ ] g If g diffil qio [ ] L whih dsibs h solios o g d µ,, h w hv zo-ip ssm, o homogos Pg of

4 MATHdo Fo mpl, osid os Fis solv h ospodig homogos qio d o fid Si µ, C C Th h gl solio o µ os loos li A Th A A, so A A A os A os A ( os d Ths, [ ( d C] os ASYMPTOTIC BEHAVIOR OF SOLUTIONS Rll fom lls h f d g smpos of o oh if lim ( f g Empl d d smpoi o h oh SEPARATION OF VARIABLES Id A diffil (o ssil li m pp s g d f d d f d g Th, ( d f ( d g, d h solio is No Oh ws spbl diffil qio pp s: M N o g d d d f d Empl d So d d d d C d No Th solio bo is implii solio Wh wi solio pliil, b fl! P io o h domi d g Pg of

5 MATHdo Empl d d d d Now, sppos ( > C W d > > >, <, h C C So IMPLICIT VS EXPLICIT SOLUTIONS Fo spbl (o li i gl, w hv implii solios (o ssil fios So s h solios o o-li qios implii, h lsis of h solio is v diffil, d w d o ow if sh solios hv plii foms o o (d wh W d o ow ivl o whih plii solios iss Empl d Cosid Th solio is d Now, if (, h d h solio is A plii solio iss o h ivl (, This solio b dd o < bs i dos pss h vil li s d A h poi (,, is dfid This idis h possibili of poblm wih dfiig d plii solio o h diffil qio Thom: Implii Fio Thom If (, b, b F d (, is sh h F d if F (, b f o ivl oiig (, b, h w hv plii fio d Colsio: O ivl s log s is dfid, w will hv plii solio d Empl d Solv sig spio of vibls d d dv L v v v d d Pg of

6 MATHdo d So d v dv d v boms v dv d v v dv d v v v ( v v So w hv w L w v, dv dw So dw d dw dw C w w l w l w l C l v l C l l C w v l l C l l l C l C O hiqs (igio limis s o v w, b w ldig som solios? Now d v ms Noi h if, d, so d w dd o h fmil of solios ISSUES ON MODELING Empl: Mo Gowh If w s h l of is is %, w m h o $, w g $ i o So P P P dp P So P A P P Eqivll, P d P P ( So ( P P P!! P P ( P P P >! If w s h l is is ompodd smi-ll, w m P P P P P So, P P P, so P P I gl, wh w hv l is ompodig ims, P( Compodig oiosl ms So P( lim P P Simill, P( P dp If w oib oiosl o h o, h P wh is h os oibio d EXISTENCE OF A UNIQUE SOLUTION Pg 6 of

7 MATHdo Empl Rll h diffil qio b bil sig fom h solio li Th boh b pssd b d Noi h h diffil qio (, is ofsd iiil vl poblm No, h solio is Rll h giv fis od diffil qio p g [ g d C] µ µ wh p d µ So s log s p d fios, h w hv good solio g igbl (oios Thom If p d g oios o I [ α, β ] solio φ h o I i sisfis p g,, h fo vl (h is ld giv, h is iq wh I Thom If h is op widow I J o whih f d L f (, poblm f (,, hs iq solio (fo I d J f oios, h h iiil vl ( I J, No Noi h is o oios ighbohood of (,, so h hom dos ppl Empl Fo whih iiil vls dos ( f (, f is oios vwh f ( hv iq solio? is lso oios vwh So ( hs iq solio fo ll iiil vls Empl Fo whih iiil vls dos f (, f is disoios d π f s ( ( o o hv iq solio? is lso disoios d π Pg 7 of

8 MATHdo BERNOULLI Bolli solvd p g s follows: p g p g v dv, d v ( v d So ow w g v p v g L AUTONOMOUS DIFFERENTIAL EQUATIONS Aoomos diffil qios loo li f Empls ; ; poil gowh is ow s logisi gowh Empl: Logisi Gowh Modl Cosid h spd of diss If is h mb of ifd poplio, h ( d (, is vioml ig pi o sio lvl d is d h iisi gowh d To solv i, d d d l l C l C A A A A A A A A A A As, A Now, A A d d ms A So So A A Empl Pg 8 of

9 MATHdo d l l d A impo modl: ( M ( l l Empl: Logisi Gowh Wih Thshold Som ims w d ogh of o s h pidmi L ( ( T f, > pois,, T Now w g d d T d d, wih iil CRITICAL POINTS OR EQUILIBRIUMS OF AN AUTONOMOUS DIFFERENTIAL EQUATION Empl qilibims givs os solios d Ths h iil pois o h smi-sbl qilibim sbl qilibim Empl givs os solios d smpoill sbl qilibim Empl: Shf Modl Pg 9 of

10 MATHdo L d b h fish poplio im, whih follows h logisi gowh modl Th E, d wh E h of hvs whih is popoiol o h fish poplio Rwiig qilibims is ( E d E, h d d E ( E Now, if > E, h sbl qilibim B if < E, h sbl qilibim is PARAMETRIC DIFFERENTIAL EQUATIONS Empl Cosid ( Th qilibims d Wh >, w hv sbl qilibim od Wh, d w hv smi-sbl solio od Wh <, w hv sbl solio od H, is lld bifio poi EXACT DIFFERENTIAL EQUATIONS Sppos ψ (, I impliil dfis fio d d ψ ψ d d Now ψ (, : M (, N(, So (, N(, d M (, d N(, d d d M o d This p of diffil qios lld qios Eis of Solio, d N, d If w giv M, how wold w ow h is sh (, M N Idd h odiio ψ ospodig o i? ψ is ss d sffii fo h is of sh (, Empl M N Cosid 6 d ( l d M (, N(, Si d, h qio is ψ Noi h M (, 6 Igig wih sp o, w g ψ (, l f ψ Now, N(, l f l, so f f H ψ (, l Th solio is (, C l ψ Pg of

11 MATHdo INTEGRATING FACTOR Wh if M N? Dos h m ψ (, dos iss? Ys, b w m b bl o hg h diffil qio M (, d N(, d o w b o If w mlipl M (, d N(, d b µ (,, w m g ( µ M ( µ N How do w ow if w shold loo fo sh µ? Fidig µ is v diffil, lss µ is somhow µ o µ ol Empl µ M d µ N d W h if i is, d if i is, h f shold ld s o h sw Sppos To b, w d Now if µ ( µ µ M µ N µ M µ M µ N µ N,, h M µ N µ ( N M µ ( N M µ Th µ N µ ( N M µ µ N Empl This lls s iio fo fidig igig fos of diff p: µ If R h iss µ N M M N, h Empl 6 Cosid d I is obvios M N, b d 6 6 N M 6 M N ms his diffil qio So h is µ h No W w looig fo igig fo µ so h µ M µn is, i ( M ( µ N ms µ M µ M µ N µ N B si w qiig h µ This µ is fio of, h b lig dµ w w, µ boms µ ( w, d µ µ d µ µ Thfo h odiio of ss dw µ N M boms µ M µ M µ N µ N µ ( M N µ ( N M R( w R µ M N Now, w sil solv fo µ NUMERICAL APPROXIMATION SOLUTIONS TO DIFFERENTIAL EQUATIONS Pg of

12 MATHdo If g is solio o h diffil qio f (, d w d o ow g ppoim vl is good ogh, h w s h g li isd of g ( ( f (, ( El sggsd o vl f (, (, f (, ( f ( (,, So, o ll g ( T, w [,T ] isd of sig f (, (,,, T This w, h sw is lo los, d d sbdivid i io Covg of El s Mhod If w l h (sp siz, h h ppoim sw qls h l sw So fo fis od diffil qios, i is good id o l h o g b solio Empl Pov h El s mhod ovgs fo, W ow h solio is L h Now, (, h h, h, h h h h h h, so h If, h ( h EXISTENCE AND UNIQUENESS OF A SOLUTION TO AN INITIAL VALUE PROBLEM Thiq A iiil vl poblm f (,, b sfomd io g(, w, w w Empl Cosid, L s ( s z, z ( z, h z( s, so h poblm boms z Now l w z z w, h z w w w s w, so Now, Thom d if f d If f (, f oios o h gl solio of h fom, h < < h φ I J, h fo som h > h is iq No Pg of

13 MATHdo To pov his hom, w fis showd i is ligh o ssm d, so w ssm h o IVP is of h fom f (,, Th mhod of h poof is Pid s Mhod No Obsv h φ is solio o f (,, So i lso sisfis φ f ( φ, φ φ ( s ds f ( φ( s, s Noi h φ φ ( s ds, i ( ds So h IVP is qivl o igl qio d Pid s Mhod Fid fio φ h sisfis φ (: φ Dfi φ sh h f ( ( s, ds,,, φ φ So w sq of { φ,,, } Now: Is his ifii sq? If som poi w sop podig w fios wh, h w ld hv h solio φ of h igl qio If h sq is ifii, h dos i ovg? To ovg, w d g φ φ ovg, i lim, [ h h], o lim φ φ, [ h h] g,, o Sppos { φ,,, } is dfid, d φ lim φ iss, h wh popis dos φ φ ( lim lim φ hv? f ( φ ( s s ds f lim φ ( s, sds lim f ( φ ( s, s ds lim f ( φ ( s, s ds lim φ φ, φ is solio So Sps of h Poof of h Eis d Uiqss Thom φ,,, wih h hops h h sq ovgs o h W osd sq of fios { } solio φ W sw h φ lim sisfis h ii fo h solio f Wh dos his sq ovg? If is d is oios i D, h h is mb K sh h f ( f (, (, f (, f, K K f (, f (, φ K φ φ Bs ( φ f, is oios o D, h fo som h, fo ll h < < h f ( s ( s ds M φ,φ, d bs φ, φ M φ, f ( M Also,, Th Pg of

14 MATHdo φ φ f ( s, φ ( s f ( s, φ ( s ds K φ ( s φ ( s ds ( MKs MK φ φ (! φ φ ds MK So, Now, oi h ( φ φ ( φ φ ( φ φ iqli, φ φ φ φ φ Fill, φ φ φ φ φ φ φ ovgs o, so b h igl ( Kh M M MK Kh K! M Kh M, so ll h φ bodd b Kh K K W d o pov h iqss of his solio Sppos h wo solios, φ d ψ φ U ψ f ( s, φ( s f ( s, ψ ( s ds f ( s, φ( s f ( s, ψ ( s ds K φ( s ψ ( s ds Now l φ ( s ψ ( s ds, wh U Noi U φ ψ K φ( s ψ ( s ds KU ( U U KU U KU U B U U φ ψ s, hfo, U ( s As Th So ds U U, i Fis Od Diff Eqios Empls of fmos dis poss sohsi posss f (,,, b,, Howv, w ol sd simpl os f (, Comp i wih f (, f (, g(, g(, ( ( Noi h A solio o diff qio is sq of mbs,, h sisfis f (, Noi h h sq dfis fio wih domi {,,, } So,,, Noi h if h ims bom smll, h diff qio boms diffil qio Empl Solv wih sp o,, So, ( ( ( ( ( ( Pg of

15 MATHdo Dfiiios W s h l is ompodig mohl o m h piod is o mohl d h o h piod is Effiv l qivl o ompodig mohl is h (ompodig ll sh h h ff of ov is h sm s h ff of fo piods Th is, if Y ( d, h No Y To ll h ffiv l, AUTONOMOUS DIFFERENCE EQUATIONS Th simpls of diff qios is oomos li f ( wh f ( b ρ Solio To solv oomos li diff qio, o h: f ( ρ b ρ ρ ρ ρ ρ f ρ ρ ρb b b ρ ρ b ρb f ( b ( b b b b b So ρ ρ b ρb b ρ b( ρ ρ ρ b ρ No Noi h if ρ <, h s, ρ ρ b b Howv, if ρ >, ρ >, s, ρ ρ ρ ρ b b ; b if ρ >, ρ <, w ofsd (h limi dos is! ρ ρ Empl Cosid mogg % (ompodd smi-ll wih mohl pm of b Wh will h mimm lo b d h hs siios? Modl: ( i A, 6 Wh is h piodi i? ( i ( i 86 Pg of

16 MATHdo So ( i i b ( i ( ( 86 7 So h mimm lo is $7 ( i i b ( i LOGISTIC GROWTH EQUATION Logisi gowh qios hv h fom ( ρ Eqilibim Eqilibim solios obid wh ρ, i ( ρ ρ ρ ρ ( ρ ρ ρ So diff vls of ρ ps diff ρ qilibims: if ρ, h h wo solios h sm; if ρ, h h wo qilibims v los Sbili of h Eqilibims Sppos ssm ss h qilibim, d sppos fo simplii h i is li, i ρ If (v h qilibim, h w m ssm logisi gowh wold b ρ ρ Th is los o, so w ssm ρ W ow s, ρ, so if ρ < Thfo h logisi gowh diff qio ρ ( will ovg o (h qilibim if w s d ρ < Mo diffil lsis gs his siio is fo oli os Th bhvio h ρ ρ qilibim L v b mo of pbio h dfis w solio ρ ρ, h is v, v This is solio d ds o b fomld o poss Noi h ρ ρ ρ v ρ ρ ρ, v ρ ρ, d ( ρ, so v ρ ( v ( ρ v ρv poss ovgs if ρ < < ρ < ρ v ρ Noi h si v, v ( ρ v, so his w Sod Od Diffil Eqios A sod od diffil qio is of h fom f (,, Pg 6 of

17 MATHdo No Bs of, i f, w hv wo dgs of fdom:, w hv ( ( Empl Cosid Solios o his si, os,, h ifiil m Spifig solios, w hv si, si, si, h sill ifii mb of To spif o solio, w d o fi lso If, h h solio is si No Somims, w poi o solio b spifig wo pois o h pl If, π, h si is h solio This is bod vl poblm If,, h his is iiil vl poblm Empl, m hv o solios, b, hv ifiil m solios So ( π ( π BVP o gd o giv iq solio, b IVP will SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS Dfiiios,, o A sod od li diffil qio os wh f ( g p q p q g If p d q oss, wh w hv os offii sod od li diffil qio No Noi h if is solio o b, h b b So if is solio o b (hisi qio of h diffil qio, h sl is solio o b If h solio o b is imgi, h os i si i Pg 7 of

18 MATHdo Pg 8 of No Noi h bs h ip is, li ombiio of d is lso solio, i is h gl solio o h diffil qio Si, so b b b b Empl Th sod od diffil qio hs h hisi qio wih oos d gl solio (h solio sp, si ll possibl vos i vo sp b wi s li ombiio of bsis vos To solv h IVP,, w solv Th gmd mi is, so b Cm s Rl, d Empl Th solio o is W lz h bhvio of h fmil, : wh h solios posiiv,, giv; wh h m/mi pois; h bhvio s ± To sw hs qsios, i is b o fo To fid, l l As, As, EQUILIBRIUM SOLUTIONS Eqilibim solios h os solios, i Empl Th qilibim solio o is

19 MATHdo W sfom his o-homogos diffil qio o homogos o L Y Y Y Y Y This sfomio is shif Y Empl v Cosid To sfom i, l Th v v Now solv v d v d f f, h f d Empl d ( is o ivolvd L v, h d Cosid dv So w hv v v d d d Now is idpd, i [ v ] d d dv dv d dv v d d d d d d FUNDAMENTAL SOLUTIONS OF LINEAR HOMOGENEOUS EQUATIONS Solios o sod od diffil qio l vld fios whih ls wi diffibl Th ollio of sh fios is dod b C W ow fom li lgb h mlipliio ( f αf α fom vo sp C d ddiio of fios ( f g f g d sl Li Tsfomios D f, D ( f D( D( f f p D f p p D f g p f g p f p g p D f p D g Th diviv f f, q : R R, q q q f q f No b A li diffil qio b pd s li sfomio So p q wi s L [ ] ( D p D q D p D q No Noi h if d Poof: L L [ ] [ ] L solios o [ ] L, h so is [ ] L[ ] L[ ] Pg 9 of

20 MATHdo Thom: Eis d Uiqss Thom ( Th iiil vl poblm L [ ] g, hs iq solio o ivl I o whih I ( g ll oios d p, q, Empl Dmi h lgs ivl I o whih h IVP ( ((, hs iq solio A,, so w hv o solio Si π No h is disoios d, h IVP is ow ( π So, π is h lgs ivl o whih w hv iq solio LINEAR INDEPENDENCE AND THE WRONSKIAN Rll h i h s of fis od diffil qio, whv w fod solio, w ddd os o i i od o ildd ll possibl solios (h gl/fmil of solios I h s of sod od diffil qios, w fid wo solios d d lim h is h gl solio How do w did if wo fios lil idpd o ivl I? Us Wosi of h wo fios Wosi If h Wosi W ( is o zo fo som I, h d lil idpd, Thom: Abl s Thom If d solios o L [ ] p q wh p d ivl I, h h Wosi ( W ( d b o o q oios o op p( d, wh is os h dpds o COMPLEX ROOTS OF THE CHARACTERISTIC EQUATION Rll: To fid solios o b w didd o fid h oos d of b, d, Pg of

21 MATHdo Wh if (pd oo, o h o l oos (ompl oos? b ± b ± b ± b i If b <, h ± λ ± iµ λ iµ λ iµ λ [ os( µ i si( µ ] As sl w s h solios ( λiµ λ iµ λ [ os( µ i si( µ ] Rl-Vld Solios B w looig fo l solio Idd, i λ os( µ λ i si( µ λ λ λ gl solio is ( µ si( µ [ os( µ si( µ ] os b If λ i b, h h solio boms os( µ si( µ os( µ si µ B lig si α α os osα solios oo So h si α α, w g os α os( µ si( µ [( si α ( os( µ ( osα ( si( µ ] Asi( µ α No Giv p q w hg h vibl o so d d d d d d d d d d d d d d d d d d d d d d d d b h p q boms Now, d d d, d d d I s o h if w l q d, Empl, o Cosid ( So q Now l d d q d d, d so, Now, boms d d ( REPEATED ROOTS b ± b Rll h If b, h So ol o solio iss: v, h mb w om p wih som oh solios s wll w l b B if Pg of

22 MATHdo is o sisf b, h: If v v v v v v v v v v d b boms ( v v v b( v v ( v [ v v bv ] v[ b ] v v bv b b b v bv bv v v v b ( So oh solio is b b b b So h gl solio is C C ( ( C C ( b NON-HOMOGENOUS DIFFERENTIAL EQUATIONS A o-homogos diffil qio hs h fom L [ ] p q g [ ] of h ssm L dids h Gl Solio Sppos d wo solios o L [ ] g, i L [ ] g L[ ] Th L [ ] L[ ] g g, so is solio o L [ ] So if is h gl solio o L [ ], h fo som d Thfo,, whih ms if o solio o L [ ] g is ow ( p pil solio h h gl solio o L [ ] g is p So o fid h gls solio o L [ ] g, w d o fis gss p h solv [ ] L How o Gss Pil Solio? Noi h (li diffil qio dos o ompll disfig h ip sobl ips will b ogizbl Rsobl ips: poil, polomil si igoomi fios si ( α, os ( α, ( α α, oos os( α g ( ls h p, Empl L si Cosid [ ] Gss p p Asi( B os L p Aos B si(, h L[ p ] si( p Asi B os p [ A os B si( ] si p A os B si( si p p Pg of

23 MATHdo p A os B si( si Asi B os A os B si si( ( A B si ( A B os si( Si si d os ( lil idpd, A A B 7 A B B 7 So gss p [ si os ] 7 Mo Complid Ips g g g g? I b dd o fidig pil solios o Wh if L [ ], b g L[ ] g L[ ] g bs L [ ] L[ ] L[ ] L[ ] g g g g L[ ] g p Empl p Cosid L p p bs p T plig A A A is ld solio o [ ] p wih p p L A A! This poblm hpps Rpd Roos If L [ ] p q g g p p [ p p p ] p[ p p ] q [ p ] g p p q p g Now solv fo p, h L[ p ] ( ( p q( p g [ ] p g p p p p p p METHOD OF UNDETERMINED COEFFICIENTS Empl Solv h IVP, Solv ± i, so h fdml solio is si os Pg of

24 MATHdo p, l p p Loo fo pil solio p Si g p p [ A D ] [ A B C D ] A A B D A D B C D A A B B A B ( A C D A C C 8 D D p os si 8 Th gl solio is Now, os si( si ( os( So So h solio o h IVP is, so 7 9 os si 8 Empl Cosid si W ld ow h fdml solios os si wh w gss h pil solio, s p A si B os os os( si So Th h gl solio is VARIATION OF PARAMETERS This hiq wos fo gl li diffil qios wih bi ip g Id If, fdml solios o L [ ] p q L [ ] g b W w mo siios o d If So g, h (, h h gl solio o Now, w solv fo d Empl Pg of

25 MATHdo Cosid l, d l l L ( l l fdml solios Rwi s l l l l Obsvio g g To solv, s Cm s l So d g W (, g g Ths, h gl solio is W (, s g s ds W (, ( s ( s g( s ds, s W REDUCTION OF ORDER If ol is giv s solio o L [ ] p q, w lim h v solio o L [ ] g So v v p v g o v ( p v g, whih w solv! is SPRING VIBRATIONS Hoo s Lw Th mo h spig shs, h mo fo F s L Eqilibim mg L mg L Empl Giv spig, g mss is hgd o i Th spig shs b m Wh is? mg ( 98 mg L 9 L Pg of

26 MATHdo Gl Solio O h w dow, h mss psss hogh im So m( g ( L m mg L m Thfo, poi (wih sp o f poi, w hv m s h diffil qio fom of Hoo s Lw λ Now,, so m µ m Th solio is ( ( ( A ( δ os si os π m (A is h mplid, δ is h phs Th piod is T π Flid Rsis Th flid h h mss is movig i ss sis λ boms m λ So ± m λ m m If λ, w hv dmpd moio wih fq If λ < m, h λ λ ± i m m m So F ~ v, so F λ So ow, h diffil qio λ ± m m A λ m λ λ q is h qsi-fq < m m m os λ m m m λ m δ, wh Chg i Mss λ λ q, > As m iss, m q iss As m dss, m m m d h will vll b o osillio λ m iss fs h m, Pdlm g Th moio of h pdlm is govd b θ θ L g is h mb of osillio p sod L π L T π is h piod g θ L ELECTRIC CIRCUITS Compos Pg 6 of

27 MATHdo dq dq Rsiso: I V VR R R d, R is sis i ohms d Cpio: V L Q, C is pi i Fd C di d Q Coil: V L L L, L is i H d d LC Cii V C V L, V L LQ, V C Q C So L Q Q wih C LC LCR Cii V C VR VL So L Q RQ Q C FORCED VIBRATION Vibio wih ip is m F γ Css of Is F F os wh Th A os( δ R os( δ, wh F F F R Si, so if h R m m m γ m h R F F whih ppohs s ppohs ifii m γ m γ F F os( wh ( γ Th A ( δ R os( δ F R Si, so m so F F os( wh γ d, h w hv so os, wh F ; if >> F F R ; if γ h R is lg (mom γ γ F si If m Th os( si( ( Pg 7 of

28 MATHdo Pg 8 of F F os wh γ d Th m F os si os If m F, h [ ] m F m F si si os os W l h l fq o b, d h h mplid ps hgig wih im (wih fq < Sis Solio of Sod Od Li Eqios Dfiiio: Rgl Poi A poi is sid o b gl poi of h diffil qio R Q P if P Th is, if P Q d P R oios, h h IVP hs solio dfid Empl Cosid h IVP, Assmig iss, w hv Th d So boms Mig h pows loo li, w hv So h DE boms [ ] [ ] [ ] 6

29 MATHdo 6 ( ( ( h ( ( ( (, so ( ( ( (,,, 6 ( ( ( ( ( ( (,,, d So h solio is Noi Now, if, h,,, d 6 So 6 Noi h if Thfo, h gl solio is b b b b, h Dfiiio: Rgl Poi A gl poi is poi wh ll h divivs of h solio fio dfid I oh wods, h Tlo sis iss RESTRICTIONS ON POWER SERIES SOLUTION Empl: Rdis of Covg Cosid ( vios pois:,, d Fid low bod fo h dis of ovg of h giv pois Th isss is wh h offii of boms ( (, so d, w hv isss,,, Th solio shll m ss o h ivls (, (, A, h dis of ovg is ls ( ρ A, h dis of ovg is ls ( ρ A, h dis of ovg is ls ( ρ Aoh hiq fo solvig diffil qios (b sis is b ol llig fw ms of h solio Empl Pg 9 of

30 MATHdo Pg of Evl fo ms of h solio o os Rwiig, w g 6 [ ] [ ] [ ] Solv i ms of d Ssm of Fis Od Li Eqios INTRODUCTION A ssm of diffil qio is li,,,,,,,,, F F F Empl Cosid h ssm os Sh ssm is wi s os o g AX X, whih is fis od diffil qio Empl W s ssm ppoh o solv diffil qio of od Cosid L, h So w hv o AX X Empl

31 MATHdo Pg of Covsl, w o sl ssm o odi diffil qio Cosid wh W hv wh 8 APPLICATIONS IN MODELING Empl: A Dobl T Miig Poblm V V d d V V d d ρ, so w hv V V V V ρ wh So ρ ρ X X wh X Empl: Dobl Mss-Spig Poblm Th vibls d So F d d m F d d m EIGENVALUES AND EIGENVECTORS Rll L/mi L/mi L/mi L/mi V L V L ρ g/mi m m F ( F (

32 MATHdo Two solios d o diffil qio idpd o ivl I if W o I (wh I ois h iiil vls (, ( Fo ssms X AX w s solio ( X, o I if (,, W X X X X X X ( ( X, ( ( X idpd Rll If is igvl of A d ξ is h ospodig igvos, i X AX X b o X b Now, if is igvo h α, h b α A ξ ξ, h i pil Noi h w m hv wo idpd igvos fo A I his s, wo solios i wo idpd diios b fod so h if ( ξ d ( ξ idpd igvos of A ospodig o igvls d, h w hv wo idpd solios X ( ( X X X ξ d h gl solio is X ξ Pg of