NATIONAL OPEN UNIVERSITY OF NIGERIA

Transcription

1 NATIONAL OPEN UNIVERSITY OF NIGERIA SCHOOL OF SCIENCE AND TECHNOLOGY COURSE CODE: MTH - COURSE TITLE: ELEMENTARY DIFFERENTIAL EQUATIONS II

2 MTH - ELEMENTARY DIFFERENTIAL EQUATIONS II Couse Te: D. O.J. Adei Wite UNI. of Agi. Abeout D. Bole Abiol Edito NOUN D. Bole Abiol Poge Lede NOUN D S.O. Ajibol Couse Coodito NOUN NATIONAL OPEN UNIVERSITY OF NIGERIA

3 CONTENT Module : Seies Solutio of Odi Diffeetil Equtio UNIT : Seies Solutio of Diffeetil Equtio UNIT: Eule Equtio UNIT: Idiil Equtio with Diffeee of Roots- Positive Itege d Logithi se UNIT 4: Boud Vlue Pobles UNIT 5: Stu d Liouville Poble ODULE : SERIES SOLUTION OF ORDINARY DIFFERENTIAL EQUATION UNIT : SERIES SOLUTION OF DIFFERENTIAL EQUATIONS. Itodutio. Objetives. Mi Cotet.. Seies Solutio of Diffeetil Equtio. Method of fidig dius of Covegee. Odi poits d Sigul poits of the Diffeetil Equtio.. Solutio e Odi Poit 4. Colusio 5. Su 6. Tuto Med Assiget 7. Refeees/ Futhe Redigs.. Itodutio: A lge lss of odi diffeetil equtios possesses solutio epessible, ove eti itevl, i tes of powe seies. I this uit we e goig to ivestigte ethods of obtiig suh solutios... Objetives: At the ed of this uit ou should be ble to - deteie dius of ovegee of seies - ppl seies solutio ethod to solvig diffeetil equtio - deteie odi poit, d sigul poits of the diffeetil equtio

4 . MAIN CONTENT.. Seies Solutio of Odi Diffeetil Equtio A epessio of the fo A A A A... is lled the powe seies. To deteie fo wht vlues of the seies oveges we use tio test ρ A li A li T T L Whee A li A L The seies is oveget whe ρ <, diveget whe ρ >. The test fils if ρ. The seies oveges whe L < R dius of ovegee L diveges whe > R L ρ is lled the dius of ovegee i If L is zeo, the seies oveges fo ll Vlues of ii If L is ifiite, the seies oveges ol t the poit iii If L is fiite, the the seies oveges, whe < R dius of ovegee d diveges if L > L If d b Covege to f d g espetivel, fo < ρ dius of ovegee ρ >, the the followig e tue fo < ρ. itwo seies be dded d subtted te wise, d

5 f b ± g ii The seies be ultiplied d g b C C b b b... b f Whee If g, the seies f g d lthough foul fo d is oplited if f otiuous hs deivtes of ll odes fo < ρ. d f, f, f b diffeetitig the seies. Thus f f o f!! is lled the Tlo seies fo futio f t A futio f tht hs Tlo seies epsio bout f f L,the f is be oputed. With dius of ovegee ρ > is sid to be lti t. The poloil is lti t eve poit, thus sus, diffeees, poduts, quotiets eepts t the zeoes of the deoito of poloils e lti t eve poit. i Deteie the dius of ovegee of the powe seies i A iv ρ li v A ii f ρ iii ρ li li li.. Deteiig the Rdius of Covegee If we obti the Tlo seies of futio f bout poit, the the dius of ovegee of the seies is equl to the diste of the poit fo the eest sigulit.

6 Re bout hge i the ide of sutio b d p p. Odi Poits d Sigul Poits of the Diffeetil Equtios We oside the diffeetil equtio d d P Q R 4 d d we ssue tht P, Q d R e poloils if P, the is odi poit of the equtio, o Q R P, Q R P, P, Q e lti t the poit, the is the odi poit of the equtio. b If the futios P, Q d R e poloils hvig o oo ftos, the sigul poits of equtio e the poits fo whih P 5 If li Q P is fiite d li Q P is fiite The the poit is lled the REGULAR SINGULAR POINT of equtio

7 d A sigul poit of equtio tht is ot egul sigul poit is lled iegul sigul poit... Solutio Ne A Odi Poit Let us oside the equtio P Q R 6 Whee P, Q d R e poloils. is the odi poit of the equtio 6. Assuig tht is solutio of 6 d hs Tlo Seies Now we ow tht! 8 We wite P q Q R whee P, q R P P q 9 o p P q q It is tul to ssue tht, t d,, we esil lulte the oeffiiet, povided tht we ould opute ifiitel deivtives of p d q eistig t. Thus p d q ust hve soe oditio fo lie lultio of. It hs bee poved tht. 7 Q P, R R P q e lti t., the the geel solutio of 6 is Whee d e bit d e liel idepedet seies solutios whih e lti t.

8 We shll illustte the ethod b eples. Eple. Solve the equtio 4 e the odi poit Solutio: we ssue the solutio s Substitutig these vlues i the equtio ields 4 o 4 4 o [ 4 ] 5 Beuse the fist two tes of the fist su i 4 e zeo. We ow use the ft tht fo powe seies to vish idetill ove itevl, eh oeffiiet i the seies ust be zeo Reuee eltio : o 4 4, Now we lulte i oeffiiets 4 4, 4, 4, , Fo bove we hve ! 4 4,!! Hee we wite i solutio

9 [ ] [ ] Si Cos!!! 4! 4 Eple : Solve the equtio 4 6 e the odi poit Solutio: we ssue the solutio The ol sigul poits of the equtio i the fiite ple e d. Hee we show tht the solutio is vlid i < with d bit oeffiiets 4 6 o 4 5 o 4 Let us shift the ide the seod seies. I equtio, the oeffiiet of eh powe of ust the zeo., fo is lled euee eltio. A euee eltio is speil id of diffeee equtio.,4,6,... d,...,5,7 4,

10 .. Siill, Hee the solutio [ ] [ ] Eple. Solve the equtio 4 bout the odi poit Solutio: we ssue the solutio We fist tslte the es, puttig u, d d d du du d. d d du d. The equtio beoes 4 u du d u du d The we ssue the solutio u 4 u u u Colletig the tes 4 u u Shiftig the ide fo to i the seod seies

11 u 7 u Theefoe d e bit d fo eide, we hve 7 bit bit ,, : [ ] [ ][ ] 6 [ ] u [ ][ ] 4 [ u u 4 Now substitute u [ [ ] 4 u u u [ ][ ] 4 4. Colusio: I this uit we hve ttepted the seies solutio ethod to Odi Diffeetil Equtios. I the subsequet uit we e goig to disuss oe bout this ethod i gete detils. You e supposed to ste this uit popel to be well equipped fo the et uit. 5. Su: Rell tht i this uit we disuss powe seies d dius of ovegee fo the seies. We lso pplied the seies to solve diffeetil equtios. We deived the sigul d odi poits fo eh of the seies solutios. Stud this uit popel befoe goig to the et uit.

12 6.. Tuto Med Assigets. Deteie lowe boud fo the dius of ovegee of seies solutio bout eh give poit fo eh of the followig diffeetil equtios. i 4, 4, 4 d ii 4,, d. Deteie whethe eh of the poits, d is odi poit, o egul sigul poit o iegul sigul poit fo the followig diffeetil equtio, 4 i ii 7. REFERENCES/FURTHER READINGS EARL. A. CODDINGTON: A Itodutio to Odi Diffeetil Equtios. Petie-Hll of Idi FRANCIS B. HILDEBRAND: Adved Clulus fo Applitios, Petie-Hll, New Jese EINAR HILLE: Letues o Odi Diffeetil Equtios, Addiso Wesle Publishig Cop, Lodo.

13 UNIT : EULER EQUATION.. Itodutio. Objetives. Mi Cotet.. Eule Equtio. Seies solutio e egul poit. Idiil equtio with equl oots.4 Idiil equtio with diffeee of oots, positive itege d No- Logithi se. 4. Colusio 5. Su 6. Tuto Med Assiget 7. Refeees/ Futhe Redigs. Itodutio: I this uit we del with lss of diffeetil equtio oll efe to s Eule Equtio. This tpe of equtio usull possesses solutios tht e lssified s egul sigul poits of the diffeetil equtios. Seies solutio of this lss of equtio ust be ttepted with diffeet ppoh. We shll see this i ou tetet of this sste of equtio i this uit.. Objetive: At the ed of this uit ou should be ble to - diffeetite Eule equtios fo othes. - use seies solutio ppoh to solve these tegoies of equtios - solve pobles eltig to Eule equtio.. MAIN CONTENT.. Eule Equtio d d L α β d d is ow s Eule equtio. It is es to see tht is egul sigul poit of I itevl ot iludig the oigi, hs geel solutio of the fo., d e lie, idepedet solutio. Hee we ssue tht hs solutio of the fo

14 L α β F Whee F α β If is oot of the equtio α α uβ α α uβ 4 F Cse I α 4β >, the the oots e el d uequl d W, is o-vishig fo d >. Thus the geel solutio is > se ll α 4β, the d we hve ol oe solutio α of the diffeetil equtio. We obti the seod solutio b the ethod of edutio. We oside diffeet ppoh to obti the solutio. L F If, the L F Now F, if we diffeetite F i.e F d the set, if give F,it suggest tht L [ F ] L log log F We set, thus L log log > is the seod solutio of Thus the geel solutio is log, > Cse III α 4β <, i this se, the oot e ople, s λ iµ, λ iµ

15 Thus the geel solutio is λ iµ λ iµ λ iµ iµ [ ] λ iµ log iµ log [ e e ] λ [ os µ log si log µ ] It is lws possible to obti el vlued solutio of Eule equtio i the itevl, b ig the followig hges d d d ξ, d d d dξ i the equtio, we hve d u du ξ ξ βu, ξ ξ > dξ dξ It is obtied s bove. Sie fo > ξ fo > It follows tht we ed oe, to eple fo b i the bove solutio to obti el vlued solutio vlid i itevl ot otiig the oigi To solve the Eule equtio α β i itevl ot otiig the oigi substitute oot d of the equtio d opute the F α β If the oots e el d uequl If the oots e el d equl log If the oots e ople λ os µ log si µ log Fo Eule equtio of the fo

16 α β Chge the idepedet vible b t o o suppose the solutio o Note: The situtio fo geel seod ode diffeetil equtio with egul sigul poit is siil to tht fo Eule equtio.. Aothe ethod of obtiig the solutio of Eule Equtio α β z Solutio: We e the hge of vible e o z log d >. d d dz d d dz d dz d d dz d d dz d dz Substitutig thee vlue i the equtio d dz α β d d This is equtio with ostts oeffiiets The uili equtio is α β i If d z z e el d uequl. e e z ii If the oots e equl i.e z e. log α iii If the oots e ople λ z e os µ z sih z λ os µ log si log µ. Seies solutio e egul sigul poit Coside the equtio P Q R

17 Assue tht is egul sigul poit of es tht Q P P R d q hve fiite liits s d e ltis t fo P soe itevl bout the oigi i But be witte [ p ] [ q ] p P q q [ p......] o p p [ q q... q...] If ll the oeffiiet use zeos, eept P d q, the edues to Eule equtio, whih ws disussed peviousl. If soe of the P d q. will ot be zeo. Howeve the essetil hte of the solutio eis the se. It is tul to see the solutio of the fo of Eule Solutio the powe seies. As pt of ou poble we hve to deteie The vlues of fo whih equtio hs solutio of the fo The euee eltio fo the The dius of ovegee of the seies We shll illustte the ethod b eple Eple I: Fid the seies solutio of the equtio Solutio: is the egul sigul poit of the equtio. We ssue lie solutio

18 Diet substitutio of i give z Now we shift the ide of the seod seies i. We get l Oe oe we eso tht the totl oeffiiet of eh powe of i the left ebe of 4 ust vish. The seod sutio does stt the otibutio, util. Hee the equtio deteits d e give b is lled the idiil equtio., v Reuee eltio 7 i Te [ ]... 5] [ ][ ]

19 Oittig the ostt, we wite the ptiul solutio s z 8 Net ts is to fid the solutio oespodig to the oot. The euee eltio beoes. b b b b. b. b.5 b b b b b b b b b if b if b b, b b b 6 6 The solutio is b[ ] The geel solutio is A B Note: The oots of idiil equtio e uequl d do ot diffe b itege.idiil equtio with equl oots Eple. Solve the equtio Solutio: is egul sigul poit of i We ssue the solutio

20 Substitutig this vlue i, we hve Shiftig the ide [ ] The idiil equtio The euee eltio is 4, I whih, [... ] 5,, 6 i whih, 7 [... ] Let s wite L The of equtio 6 hs bee so deteied tht fo tht the Eight ebe of 8 edued to sigle te the. Thus L 8 A solutio of the oigil diffeetil equtio is futio fo whih L. Now tig l es L [, ] Now diffeetite eh ebe of 9 with espet to [, ] [ d l[, ] d ] log

21 Fo 9 d, it be see el tht the two solutio of the equtio L e [, ] [, d [, ],, ] d log, log log [... ] log log [log...log [... ], we obti We wite H... The solutios e! [ log H... ] ]! The geel solutio, vlid fo ll fiite log is A B.

22 .: Idiil Equtio with diffeee of Roots Positive Itege, o Logithi se Solve lie equtio 4 Solutio: We ssue the solutio L 6 The Idiil equtio is 5, 5 s 5 5 We eso tht we hope fo two powe seies solutios, oe sttig with 5 d te, oe with te. If we use the loge oot 5, the the te would eve ete. Thus we use the slle oots, the the til solutio of the fo. hs he of piig up both solutios beuse the 5 5 oti L 5 L L o o s te does

23 [ ] Theefoe, with d 5 6 [ _ [ ] 5 4 bit, the geel solutio be witte Eple Solve the equtio 4 Solutio is the egul sigul poit of. We ssue the solutio L 4 L

24 , we get the idiil equtio, L Usig the slle oot L 4 Reuee eltio is o > l t 9 6 This is the equied solutio 4.. Colusio We hve looed t vious pobles ivolvig Eule equtios i this uit thei vious fo of idiil equtios. I the et uit we shll oside idiil equtio of positive itege d logithi se.

25 5. Su You will ell i this uit tht geel fo of Eule equtio ws give. We lso oside vious fo of Eule equtios. You e equied to ste this uit ve well befoe poeedig to othe uits. 6.. Tuto Med Assiget:. Solve the equtio.solve the equtio 4 7. REFERENCES/FURTHER READINGS. EARL. A. CODDINGTON: A Itodutio to Odi Diffeetil Equtios. Petie-Hll of Idi. FRANCIS B. HILDEBRAND: Adved Clulus fo Applitios, Petie-Hll, New Jese. EINAR HILLE: Letues o Odi Diffeetil Equtios, Addiso Wesle Publishig Cop, Lodo.

26 UNIT: INDICIAL EQUATION WITH DIFFERENCE OF ROOTS A POSITIVE INTEGER, LOGARITHMIC CASE. Itodutio. Objetives. Mi Cotet.. Idiil Equtio with diffeee of oots, positive itege d logithi se. Fouie Seies. Othogolit of set of seies d osies 4. Colusio 5. Su 6. Tuto Med Assiget 7. Refeees/ Futhe Redigs.. Itodutio I uit we hve osideed idiil equtios whee logith se is ot osideed. We shll udete to oside the positive d logith ses i this uit.. Objetives At the ed of this uit ou should be ble to - to solve diffeetil equtio whose idiil equtio hs positive itege - to solve diffeetil equtio whose idiil equtio hs oots with logithi se... MAIN CONTENT.. Idiil Equtio with Diffeee of oots positive itege, logithi se. We illustte this ethod b eple Solve the equtio Solutio: We ssue the solutio L 9

27 The idiil equtio is Puttig,, the euee eltio [... ] [... ] It follows tht L o Fo, ol oe solutio be obtied. Note: fo, sie thee is o powe seies with sillig, we suspet the pesee. Choose, we hve, [... ] We obti two solutios with espet to Fo whih L [, We use the se guet s tht of equl oots Puttig, [, ], [... ] 4 Diffeetite with espet to., log 4 { } { u

28 ... [... Puttig e, get ] [..... o ] [... }... { } { log! ] ] [ log H Poble: Fid lie geel seies solutio of the D.E 6 4 d d d d d show tht it be epessed i lie fo Solutio: is egul sigul poit of D.E. Assuig the solutio Substitutig i the D.E Chgig the ide. The idiil equtio is,, The euee eltio is ] [..... o

29 , i, the, 45 Thus! Hee the solutio is h!! Si ii,, the ! os ] Hee the geel solutio is ACos BSi

30 .. Fouie seies. Othogolit: A set of futio is { f, f,... f,...} sid to be othogol set with espet to the weight futio w ove the itevl b if b w f f d fo fo Othogolit is popet widel eouteed i eti bhes of thetis. Muh use is de of the epesettio of futios i seies of the fo f I whih the e ueil oeffiiets d { f } is othogol set is. Othogolit of set of seies d osies: We shll oside the set of futio Si π,,,.... π Cos, o,,,... Si π, Si π, Si π, Si π. π, Cos, π Cos, π π Cos, Cos.. is othogol with espet to the weight futio w ove the itevl i.e. π π Si Cos d, whee. : Befoe we pove the esult, we give soe defiitio to shote the poof. Eve futio: A futio g is sid to be eve g g Fo ll. b Odd futio: A futio h is odd if h h Fo ll.

31 Eple: Si is odd futio Cos is eve futio Most futio e eithe eve o odd eple. is i oe futio f d If g is eve futio the s well s eve g d Coside the itegl. g d π π I Si Cos d fo ll d. Follows t oe fo the fts tht the itegd is odd futio of. It does ot depedet upo oe ft d d e iteges π π I Si Si d, Te π β, d dβ π π π Cos β Cos β ] dβ π λπ π Si Si β π [ β ] π π Sie d e ve iteges. Fill we oside the itegl Whee,,....., Si β Si β π [ ] π π λπ SiβSiβdβ π I 4 Si d Si d let f the otiuous d diffeetible t eve poit i itevl eept fo ost fiite ube of poits d t oe poits, let f d f hve ight d left-hd liits Note:

32 The ottio f is used to deote lie ight-hd liit of s f fo loe, i.e. f d f Siill li f f li Deotes, the liit of f s ppohes. Sie Fouie seies fo f ot ovege to the vlue f eve whee. It is usto to eple the equls sig i equtio 8 b the sbol ~whih be ed hs fo its Fouie Seies we wite π π f ~ os Whee b si, d b e give b d Eple: Costut the Fouie seies, ove the itevl, fo the futio defied b f,,, < < Solutio: Now π π f ~ os b si I whih d π f os d;,,... π b f si d;,,... π os d π os d If, the π π π [si [ si os π π π os π π ] ]

33 Fo, fo, we get b Thus we wite π π f ~ [ os si ]. π π Eple Obti the Fouie seies ove the itevl π toπ fo the futio Solutio: We ow ~ π π π π π [ os os d; b si ] fo π < < π, whee,,,... b si d;,,,... π is eve futio, si is odd futio, thus si is odd futio Hee. b. fo eve. π si os si π [ ] π π fo Fo whih π os π 4 [ ],,,... π π osπ 4 [ ],,... π fo π π π [ π d π Theefoe, i the itevl; π < < π π os ~ 4 Ideed, beuse of oditio of lie futio ivolved, we wite π os ~ 4, fo π π.

34 . Fouie Sie Seies: Soeties it is desible to epd the futio f i seies ivolvig Sie futio ol. I ode to get Sie seies fo f we itodued futio g defied s follows g f < < f < < Thus g is odd futio ove the itevl < < Hee π g ~ os d.,,,... It follows tht π g os d Note itegd is odd futio d tht,,... b Thus π g si d π f si π f ~ b si o < < π f Whee b f si d,, Eple: Epd f i Fouie Sie Seies ove the itevl < < Solutio: At oe we wite, fo < ~ I whih b b si π si πd os π si π π os π π π <

35 osπ [ π osπ π osπ π Hee the Fouie Sie Seies, ove < < fo is os π [ ] π π π. Fouie Cosie Seies: I ode to epd the futio f i seies ivolvig osie futio ol, suh seies is lled Fouie Cosie Seies. We defie h f < < f < < It follows tht h is eve futio of. π π h ~ os b si π π h d f d os os < < i π But b h si d Thus we hve f ~ π f os d i whih π f os d Eple: Solutio: At oe we hve π f ~ os i whih i whih π os d. π π [ si os ] π π

36 . [ os π π ] π os π, π The oeffiiet is edil obtied d Thus the Fouie Cosie Seies ove the itevl < < the futio f is u f π ~ os π 4.. Colusio: You hve let bout idiil equtios whee the oots e positive d logithi. You hve lso let bout Fouie seies d odd futios. 5. Su: You e equied to stud teils i this uit ve well befoe poeedig to the et uits. 6. Tuto Med Assigets:. Fid the geel seies solutio of the D.E d d 4 6 d d.costut the Fouie seies, ove the itevl, fo the futio defied b f,,, < < 7. REFERENCES/FURTHER READINGS. EARL. A. CODDINGTON: A Itodutio to Odi Diffeetil Equtios. Petie-Hll of Idi. FRANCIS B. HILDEBRAND: Adved Clulus fo Applitios, Petie-Hll, New Jese. EINAR HILLE: Letues o Odi Diffeetil Equtios, Addiso Wesle Publishig Cop, Lodo.

37 UNIT 4: BOUNDARY VALUE PROBLEMS. Itodutio. Objetives. Mi Cotet.. Boud Vlue Pobles. Eige Vlues d Eige Futios 4. Colusio 5. Su 6. Tuto Med Assiget 7. Refeees/ Futhe Redigs. Itodutio I this uit, we will disuss soe of the popeties of boud vlve pobles fo lie seod ode equtio. This lss of diffeetil equtios is ve useful fo ptil pplitios. We shll devote soe tie i studig the i this uit.. Objetives: At the ed of this uit ou should be ble to - lssif seod ode diffeetil equtios ito hoogeeous d o-hoogeeous. - diffeetite betwee eige vlues d eige futios - solve elted eige vlue pobles. MAIN CONTENT. Boud Vlue Pobles The lie diffeetil equtio P Q R g ws lssified hoogeeous if, g, d o-hoogeeous othewise. Siill, lie boud oditio A boud vlue poble is hoogeeous if both its diffeetil equtio d i boud oditios e hoogeeous. If ot the it is ohoogeeous.

38 A tpil lie hoogeeous seod ode boud vlue poble is of the fo. P Q R < <, 4 b I b 5 Most of the pobles, we will disuss e of the fo give b to 5... Eige Vlues d Eige Futios Coside the diffeetil equtio p, λ q, λ < < The boud oditios b I b Whee λ is bit pete. Clel the solutio of depeds o d λ d be witte s, λ, λ, 4 Whee d e fudetl solutio of. Substitutig fo i the boud oditio d, ield. [ o, λ o, λ] [, λ] 5 b i, λ b i, λ] [ b i, λ] 6 [ A set of two lie hoogeeous lgebi equtios fo the ostt. Suh set hs solutios othe th if d ol if the deteit of oeffiiets Dλ vishes i.e.

39 [ o, λ o, λ] o, λ o, λ D λ b i, λ b i, ] b i, λ b i, ] λ λ Vlues stisfig this deteit equtio e the eigevlues of the boud- vlue pobles, d Coespodig to Eige futio eh Eige vlue is t lest oe o- tivil solutio. A Note: We will oside pobles el ol el eige vlue Eple I oside the equtio λ, Solutio: λ, the solutio is si, os B the boud oditios si λ is the solutio si λ λ π,,, o λ π, 4π 4 4 gives the eige vlues of. If we oside λ os si b si I b Hee λ is the eige -futio The eige futio e si π,,.... is bit ostt. 5

40 Eple : Fid the el eige vlues eige -futio of the boud vlue poble λ l.the solutio is λ os λ, give. Also os λ os λ But l, ields λ os λ l λ osl π λl, l,, gives eige vlue π si[ ], l,, gives eige futios Eples λ Solutio: The solutio is λ si λ os si λ os λ µ os µ µ os µ Thus the eige vlue e give b the equtio µ t µ. µ µ si µ If µ is the oot of, the eige futio is si µ µ os 4 µ

41 If λ, the the solutio is Hee the solutio is thus λ is lso eige vlue µ t µ λ ~ 4.49, Eple 5. Solutio λ N π ~ λ, os λ si λ, λ ot λ λ os The eige vlues e give b equtio. The eige futio e λ Whee the oot of is λ is the oot of the equtio λ ot λ λ, ot λ λ π,,..... λ π fo lge Eple 6. Coside the poble λ, Show tht if,d e eige futio oespodig to the eige vlue λ d λ Respetivel, the

42 d l Povided tht λ λ. Solutio: λ λ λ λ λ d d l l λ d d l l λ 4 d d l l l λ d d o l l l o l ] [ λ o B boud vlue oditios d d l l λ 5 Subtt 4 fo 5, we hve d l λ λ If λ λ, the d l Eple 4. Hpeboli futio os e e, sih e e sih osh d d osh si h d d

43 Solutio of the poble is 4 λ, Te λ 4 4 µ 4 µ The solutio is os µ si µ µ os µ sih µ The boud oditio d si µ 4 si µ l si µ l 4 sih µ l si µ 4 si µ l si µ l 4 sih µ l si µ 4 si µ l si µ l 4 sih µ l si µ 4 si µ l si µ l 4 sih µ l si µ l sih µ l,, si µ l π π si,, l Is the eige-futio 4. Colusio We hve bee ble to stud soe eige-vlue pobles i this uit. This uit ust be steed popel befoe ovig to the et uit. 5. Su Rell tht the lie diffeetil equtio P Q R g ws lssified hoogeeous if, g, d o-hoogeeous othewise. Siill, lie boud oditio

44 A boud vlue poble is hoogeeous if both its diffeetil equtio d i boud oditios e hoogeeous. If ot the it is ohoogeeous. We lso lssified soe equtios ito eige vlue poble depedig upo whethe the deteit of the eige vlue of the poble is zeo o ot. Red efull d e wo ll eeises d pobles i this uit fo bette udestdig. 6. Tuto Med Assiget:. Coside the poble λ, Show tht if,d e eige futio oespodig to the eige vlue λ d λ Respetivel, the l d Povided tht λ λ.. Fid the el eige- vlues d eige -futio of the boud vlue poble λ l 7. REFERENCES/FURTHER READINGS. EARL. A. CODDINGTON: A Itodutio to Odi Diffeetil Equtios. Petie-Hll of Idi. FRANCIS B. HILDEBRAND: Adved Clulus fo Applitios, Petie-Hll, New Jese. EINAR HILLE: Letues o Odi Diffeetil Equtios, Addiso Wesle Publishig Cop, Lodo.

45 Module : Stu Liouville Boud Vlue Pobles d Speil Futios UNIT: Stu d Liouville Poble. Itodutio. Objetives. Mi Cotet.. Stu d Lioville Pobles 4. Colusio 5. Su 6. Tuto Med Assiget 7. Refeees/ Futhe Redigs.. Itodutio We solved soe ptil diffeetil equtios b the ethod of septio of vibles. I the lst step we epded eti futio i Fouie seies, i.e. s the su of ifiite seies of sie d osie futios. It is of fudetl ipote tht the eige futios of oe geel lss of boud vlues pobles be used s bsis fo seies epsios, whih hve popeties siil to Fouie Seies. Suh eige- futios seies e useful i etedig the ethod of septio of vlues to lge lss of pobles i ptil diffeetil equtio. The lss of boud vlue poble we will disuss is ssoited with the es of Stu d Liouville.. Objetives: Afte studig this uit ou should be ble - to solve ptil diffeetil equtio usig Stu d Liouville ethods - solve oetl the ssoited Tuto Med Assigets. MAIN CONTENT.. Stu d Liouville Poble We itodue the opeto L [ ] [ p ] q L [ ] λ

46 [ P ] q λ o the itevl < <, togethe with the boud oditio 4 b I b 5 We shll ssue tht p, q d e otiuous futios i the itevl [,]. P >, > fo ll i. i Lgge s idetit: let u d v be futios hvig otiuous seod deivtives o the itevl. The UL [ U ] UL[ u] d p [ u u u u ] 6 Solutio : UL [ U ] d { u[ p u q u} d u p u p u { u pu uqu} d ul [ u] UL[ U ] d p[ u u u u This is ow s Lgge s idetit if u d u stisf 5 d 4 R.H.S p [ u u u u ] p [ u u u u ] b b p [ u u u u] b b Thus we hve p [ u u { u [ u] ul[ u] d u u] ii Show tht ll the eige vlue of the Stu-Liouville poble

47 L λ A With boud oditios B b I b e el. Poof: let us suppose thee eists ople eige vlue λ µ iv will v d oespodig to this vlue is the eige futio Q U IV Whee t lest oe of the is ot idetill zeo. Now Q stisfies the diffeetil equtio L[ Q λq L[ Q] πrq o u Q d u Q { QL Q QL Q] d Q Q d λ λ o [ V V d. Sie > fo ll i v This otdits the oigil hpothesis. Hee the eige vlue of Stu- Liouville poble e el. iii If Q d Q e eigevlues of the Stu-Liouville poble A d B, oespodig to eige vlves λ d λ, espetivel, d λ λ, the Q Q d [ is lled the weight futio d it is othogol popet of eigefutio] Poof: - L[ Q ] λ Q

48 L[ Q ] λ Q If we let U Q d U Q { ul [ U ] UL[ U ]} d λ λ Q Q d the Hee the esult iv Let us ow oside oe geel boud vlue poble fo the diffeetil equtio L[ ] λm[ ], < < Whee L d M e lie hoogeeous diffeetil opetios of odes d espetivel. L[ ] p p... p p M[ ]... Whee >. I dditio to the diffeetil equtio set of lie hoogeeous boud oditios t, is lso pesibed. If the eltios ul [ u] ul[ u] d um [ u] um[ u] d e lie fo eve pi of futios u d u, whih e lies otiuousl diffeetible o, l d whih stisf u give boud oditios, the the give boud vlve poble is sid to be self djoit. Poble I. Show tht the Stu-Liouville pobles L [ P ] q M λ i [ UM [ u] um[ u] d [ U λ u uλ u] d Fo eve pi of u, u

49 [ UL [ u] ul[ u] d s show peviousl. Hee it is self-djoit Poble, I Solutio L i [ U u u u u u u u d u ud, e tue fo eve pi of futio u d u, whih e - ties otiuousl diffeetible o [ o, l] whih stisf u give boud vlue poble is sid to be self-djoit. Solutio: Stu-Liouville pobles L [ p ] q M λ i [ UM u um u] d [ U λ uλ ] d s show peviousl. Hee it is ot self-djoit. Poble, I \ Solutio L, I [ U u u u u u u u] d [ UL u ul u] d s show pevious. Hee it self-djoit Poble Poble, I \ Solutio L, I [ U u u u u u u u] d

50 u ud, u d u e otiuous i the itevl. Hee it is ot zeo. Thus it is ot self-djoit. Pobleb, L U,, L M [ U [ u u u u] d It is Stu-Liouville poble. λ, Solutio L M i um µ UM µ d uu uu d ii u u U µ d] u u u u d u u uu [ u u u u [ u u u u ] [ u u u I u The ight side is ot zeo. Hee it is ot self-djoit. Poble Coside the diffeetil equtio λ \ With boud oditios

51 I, Show tht the poble is self-djoit eve though it is ot Stu- Liouville poble. b Fid ll eigevlues d eigefutios of the give poble Solutio: L M i [ UM u u u d] ii [ U u u u] d [ u uu uu ] d u u u u d u u u u d [ u I u I u I ] [ u u u] [ u u u] [ u u u] Hee it is self-djoit The solutio of the equtio is os si λ Applig the boud oditios, we hve λ si λ si λ os λ os λ si λ Thus λ si λ λ os λ os λ si λ O λ os λ λ o λ π,,,..... λ ϕ

52 λ π Q os π, si π, Q os π, si π, osπ, π os π, W os π π si π si π π os π π os π π si π π Betwee Thus the eige futios e liig idepedet. Poble 5: Coside the Stu-Liouville pobles [ p ] q λ, b, b Whee p, q d z otiuous futio i the itevl. show tht if λ is eige-vlue d ϕ oespodig eige futio, the b λ eq d p q d p p b Povided tht b How this esult be odified if b b Show tht if q d if. d b e o-egtive, the the eige-vlue λ Is o egtive Show tht the eige-vlue λ is stitl ude q fo eh i,, Solutio λ Q [ p Q ] Q q Q Thus

53 λ Q d qq p Q Q d Itegtig b pts, we hve qq d Q[ p Q ] pq d Fo boud oditio, we hve obti the esult b Q Q b Q Q Puttig these vlues o the ight side d we obti the esult if o o b, the the fist boud oditio edues to Q o Q o The esult edues to λ o b Q d qq pq d b p Q I λ Q d qq pq d Q Q I b I Stu Lioville poble, we lws ssue tht p >, >, B give oditio > fo ll i Q > fo ll. Now we ipose oditio, so tht ight side of the equtio i is b The seod d thid te e ve d e o-egtives b ve Now qq d is ve i ode tht

54 If q fo ll the λ is stitl. 4. Colusio We hve studied the Stu Lioville poble i this uit. You e to ste this uit popel so tht ou will be ble to solve the pobles tht follow. 5. Su Rell tht Stu Lioville pobles e usull pobles ssoited with eige vlues pobles of ptil diffeetil equtios whih we hve delt with i this uit. I ou subsequet ouse i thetis i this Poge, we will hve use to del with it gi ptiull whe will shll stud Ptil Diffeetil Equtio. 6. Tuto Med Assiget Coside the poble λ u Y, Show tht this poble is ot self-djoit Show tht ll eigevlues e el Show tht the eigefutios e ot othogol. with espet to the weight futio isig fo the oeffiiets of.. i the diffeetil equtio. 7. REFERENCES/FURTHER READINGS. EARL. A. CODDINGTON: A Itodutio to Odi Diffeetil Equtios. Petie-Hll of Idi. FRANCIS B. HILDEBRAND: Adved Clulus fo Applitios, Petie-Hll, New Jese. EINAR HILLE: Letues o Odi Diffeetil Equtios, Addiso Wesle Publishig Cop, Lodo.