Subject: Differential Equations & Mathematical Modeling -III. Lesson: Power series solutions of Differential Equations. about ordinary points

Transcription

1 Power series solutio of Differetial equatios about ordiary poits Subject: Differetial Equatios & Mathematical Modelig -III Lesso: Power series solutios of Differetial Equatios about ordiary poits Lesso Developer: Nisha Bohra ad Akit Gupta ollege/departmet: Sri Vekateswara ollege, Uiversity of Delhi Ramjas ollege, Uiversity of Delhi Istitute of Lifelog Learig, Uiversity of Delhi

2 Power series solutio of Differetial equatios about ordiary poits Table of otets. Learig Outcomes. Itroductio. Basic ocepts ad Results.. Secod Order homogeous liear differetial equatio.. Power Series ad its Radius of overgece.. Power Series method for solvig a differetial eqatio Eercises 4. Ordiary ad Sigular poits Eercises 5. Series Solutio ear Ordiary Poit Eercises 6. Legedre s Equatio Summary Refereces Istitute of Lifelog Learig, Uiversity of Delhi Page

3 Power series solutio of Differetial equatios about ordiary poits. Learig Outcomes After readig this lesso reader will be able to uderstad the followig Secod order homogeous liear differetial equatio Power Series Radius of covergece Ordiary poits, sigular poits Solutio about a ordiary poit Legedre s equatio. Itroductio We have studied earlier that the problem of solvig a homogeous liear differetial equatio with costat coefficiets ca be simplified to the algebraic problem of fidig the roots of characteristic equatio. However there is o parallel system for solvig liear differetial equatio with variable coefficiets. Thus we must seek other techiques for the solutios of these equatios ad this chapter is devoted to the methods of obtaiig solutios i ifiite series form.. Basic ocepts ad Results.. Secod order homogeous liear differetial equatio A differetial equatio of the form A B y, where the coefficiets A, B ad are cotiuous real fuctios of o a iterval I ad A() is ot idetically zero o I, is called a secod order homogeous liear differetial equatio... Power Series ad its radius of covergece Defiitio. A power series about a is a ifiite series of the form a a a... (.) where s are costats. For the sake of simplicity of our otatio, we shall treat oly the case whe a=. ' a reduces a power This is o loss of geerality, sice the traslatio series aroud the poit stu a series of the form to a power series aroud zero. Thus, we shall maily... (.) Istitute of Lifelog Learig, Uiversity of Delhi Page

4 Power series solutio of Differetial equatios about ordiary poits Eve though the series (.) is defied over all of R, it is ot to be epected that the series will coverge for all i R. For eample, the geometric series... coverges for ad the series for all i R as ca be see by Ratio test. Radius of overgece (R) of a power series /! coverges Defiitio. For a give power series R, by, we defie the umber R, R lim sup /, the a) If b) If c) If R, the series coverges absolutely. R, the series diverges. r R, the the series coverges uiformly o : r. The umber R is called the radius of covergece ad the ope iterval RR, is kow as iterval of covergece of the power series. Remark. The radius of covergece R of the series lim N is also give by, provided this limit eists. If R, we say series coverges for all ad if R, the series diverges for all. Diverges overges absolutely Diverges Eample. If y the R lim Istitute of Lifelog Learig, Uiversity of Delhi Page 4

5 Power series solutio of Differetial equatios about ordiary poits lim lim Hece the series coverges for ad diverges for. We ow state some basic results o Power series without proof. Theorem. Term wise differetiatio of Power series Let a fuctio f has the followig power series represetatio, f If it coverges o the ope iterval I, the we say, f is differetiable o I ad f... at each poit of I. Both series have the same radius of covergece. Theorem. Idetity priciple If a b for every i a ope iterval I, the a = b for all. I particular, if a for all i I, it follows that a... Power series method for solvig a differetial equatio I this sectio, we illustrate the power series method for fidig the solutio of a differetial equatio. Give a differetial equatio, we will assume that it has a solutio of the form y( ). The, by substitutig the above power series i the give differetial equatio, we will fid the coefficiets,,,. or the set of coditios which must be satisfied by them. Let us try to uderstad this method with the help of a eample. Eample. Solve the equatio y. (.) Istitute of Lifelog Learig, Uiversity of Delhi Page 5

6 Power series solutio of Differetial equatios about ordiary poits Solutio. Let y( ) be the solutio of (.). Differetiatig term by term we obtai. Substitutig the values of y ad y ito the differetial equatio (.), we have Sice is idepedet of the ide of summatio, we may rewrite this as. I order to make the epoet of same i all the three summatios, we shall rewrite secod summatio. osider the secod summatio. To make the epoet, we first replace the preset epoet (-) by a ew variable m. That is, we let m = -. The = m+, ad sice m= for =, the m summatio takes the form m ( m ) m. Now sice the variable of summatio is merely a dummy variable, we may replace m by to write the secod summatio as ( ). (.4) Replacig secod summatio by its equivalet form (.4), we get.. The idetity priciple gives,. This yields the recurrece relatio for. Put =,,, we get Istitute of Lifelog Learig, Uiversity of Delhi Page 6

7 Power series solutio of Differetial equatios about ordiary poits 4,,. Geeralizig the above patter, we ca write, Hece power series solutio of (.) is y. (.5) The radius of covergece of above power series is R lim. Hece (.5) coverges absolutely if ad diverges for. Eercise Fid the radius of covergece of followig power series ! ( ) ( )! ( ) Fid a power series solutio of followig differetial equatios. Also, determie the radius of covergece of the resultig solutio series.. y. y. 4y 4. y 4. Ordiary ad Sigular Poits I Eample of previous sectio, we assumed that the differetial equatio has a power series solutio. However it is ot always true that the differetial equatio possesses a solutio of this form. Hece the atural questio is uder what coditios this assumptio is actually valid? I order to aswer Istitute of Lifelog Learig, Uiversity of Delhi Page 7

8 Power series solutio of Differetial equatios about ordiary poits this importat questio cocerig the eistece of a power series solutio, we shall first itroduce a few basic defiitios. Defiitio. A real valued fuctio f defied o a iterval I cotaiig the poit a f a is said to be aalytic at = a if its Taylor series about a, a eists! f for all i some ope iterval cotaiig a. ad coverges to Value Additio: Note All polyomial fuctios, epoetial fuctio ad trigoometric fuctios like si ad cos are aalytic everywhere. A ratioal fuctio is aalytic ecept at poits where deomiator is zero. For eample, the ratioal fuctio defied by / is aalytic everywhere ecept at = ad =. osider the secod order homogeeous liear differetial equatio of the form A B y (4.) where the coefficiets A, B ad are aalytic fuctios of. Rewritig equatio (4.) i the form P Q y (4.) with leadig coefficiet ad with P B / A ad / Q A. Equatio (4.) is called equivalet ormalized form of equatio (4.). Note that P ad Q will fail to be aalytic at poits where A is zero. Defiitio. A poit = a is called a ordiary poit of the differetial equatio (4.) ad the equivalet equatio (4.) if both the fuctios P() ad Q() are aalytic at = a. If either (or both) of these fuctios are ot aalytic at = a, the a is called sigular poit. Eample. osider the differetial equatio d y y. Q. Here P ad Both fuctios P ad Q are polyomial fuctios ad so they are aalytic everywhere. Thus every poit is a ordiary poit of the give differetial equatio. Eample. osider the differetial equatio y. Istitute of Lifelog Learig, Uiversity of Delhi Page 8

9 Power series solutio of Differetial equatios about ordiary poits We first epress the above differetial equatio i ormalized form, thereby obtaiig d y y. Here, P ad Q. The fuctio P is ot aalytic at = ad Q is ot aalytic at the poits = ad. Thus = ad = are sigular poits of give differetial equatio. All other poits are ordiary poits. Eample : osider the differetial equatio si y. The the poit = is a ordiary poit because the fuctio 5 si P( )...! 5! 4...! 5! is clearly aalytic at = ad Q aalytic at =. Eercise beig a polyomial fuctio is also I problems to 4, determie the ordiary ad sigular poits. y.. 6 y.. y.. 4 y Series solutio ear ordiary poits We ow state a theorem cocerig the eistece of power series solutios of the form a. Istitute of Lifelog Learig, Uiversity of Delhi Page 9

10 Power series solutio of Differetial equatios about ordiary poits Theorem. Suppose that a is a ordiary poit of the differetial equatio A B y (5.) i.e., the fuctios P B / A ad Q / A are aalytic at = a. The equatio (5.) has two liearly idepedet solutios, each of the form y a (5.) ad these power series solutios coverges i some iterval a R about a (R > ). We shall omit the proof of this importat theorem. Remark. The radius of covergece R of such series solutio is at least as large as the distace of a from the earest sigular poit (real or comple) of the equatio (5.). Eample. Determie the guarateed radius of covergece of a series solutio of ( 9) y i powers of. Repeat for a series i powers of (-4). Solutio. We first write the differetial equatio i ormalized form. d y y ( 9) ( 9) Here P 9 ad Q give differetial equatio are.. Thus, the oly sigular poits of 9. The distace of both from is, so the radius of covergece of series solutio is at least. Sice, the distace of both sigular poit from 4 is 5, so a series solutio of the form Istitute of Lifelog Learig, Uiversity of Delhi Page

11 Power series solutio of Differetial equatios about ordiary poits ( 4) has radius of covergece at least 5. Remark. Theorem gives us oly sufficiet coditio for the eistece of power series solutios of the differetial equatio (5.). I Eample of sectio 4, we observed that = ad = are the oly sigular poits of the give differetial equatio. Thus the differetial equatio has two liearly idepedet solutios of the form (5.) about ay poit or. However we are ot assured that ay solutio of the form ay solutio of the form about the sigular poit = or about the sigular poit. Let us look at some eamples based o fidig the power series solutio of give differetial equatio about a ordiary poit. Eample. Fid the geeral solutio of the differetial equatio d y y (5.) i powers of (that is about a = ). Solutio. learly a = is a ordiary poit of (5.). We assume that y( ) is a solutio of (5.). Differetiatig term by term, we obtai ad. Substitutig these values i equatio (5.), we obtai.. To make the epoet of same i all the summatios, we shift the ide by i the first sum, replacig by ( + ) ad usig the iitial value =. This gives Also i the third sum, we replace by ( ) to get Istitute of Lifelog Learig, Uiversity of Delhi Page

12 Power series solutio of Differetial equatios about ordiary poits (5.4) Sice rage of various summatios is ot the same. The commo rage is from to. We ow write the terms i each summatio for = ad = separately ad we cotiue to use the "sigma" otatio for the remaider of each such summatio. Thus equatio (5.4) reduces to 6. This gives 6 Equatig coefficiet of each power of i the left to zero, we get 6, (5.5) & The coditio (5.5) is called recurrece formula. It eables us to epress each coefficiet for i terms of the previous coefficiets ad, thus givig, Puttig =, we get 4. 4 For =, 5 5 ad the usig the value of, we have 5 4 I this way, we ca epress each eve coefficiet i terms of ad each odd coefficiet i terms of. Substitutig the values of,, 4 & 5 i the assumed solutio, we have 4 5 y( ) Istitute of Lifelog Learig, Uiversity of Delhi Page

13 Power series solutio of Differetial equatios about ordiary poits ollectig terms i ad, we have (5.6) 4 5 y( ) which gives the solutio of the differetial equatio (5.) i powers of. The two series i the parethesis are the power series epasios of two liearly idepedets solutios of (5.) ad, are arbitrary costats. Thus (5.6) represets the geeral solutio of (5.) i powers of. Eample. Fid the geeral solutio i powers of of 4 y. (5.7) Also, fid the particular solutio usig coditios y() = 4, y '. Solutio. learly a = is a ordiary poit of (5.7). We assume that y( ) (5.8) is a solutio of (5.7). Differetiatig (5.8) term by term, we get Substitutig these values i (5.7) yields We ca chage the iitial value from = ad = to = i the first ad third summatio without affectig the sum. Also, by replacig with + ad usig the iitial value =, we shift the ide of summatio i the secod term by +. This gives Istitute of Lifelog Learig, Uiversity of Delhi Page

14 Power series solutio of Differetial equatios about ordiary poits ( ) 4. The idetity priciple yields 4,. 4, (5.9) 4 (5.9) gives the recurrece relatio for. With =, & 4, we get 4., = = Geeralizig the above patter, we have ,.! With =, & 5 i (5.9), we get 4.4, ad Geeralizig the above patter, we have ! ( ), Usig these values i the assumed solutio (5.8), we get Istitute of Lifelog Learig, Uiversity of Delhi Page 4

15 Power series solutio of Differetial equatios about ordiary poits..5...! y.! OR 4 5 y Sice y & y. Usig iitial coditios, we have = 4 ad =. Hece, the particular solutio of (5.7) is 4 5 y We ote that sigular poits of equatio (5.7) are, so the radius of covergece of above series is at least. Traslated Series Solutios. Suppose we are required to solve a differetial equatio with iitial values specified at a poit a, i.e. y a ad y a are give. To fid a particular solutio with give iitial values, we will eed a geeral solutio of the form. y a Because the y a ad y a. Ad thus we have the values of & i terms of the y a ad ' y a. Therefore, to solve a iitial value problem, we require a geeral solutio cetered at the poit at which iitial coditios are give. Let us look at some eamples based o this. Eample 4. Fid a power series solutio of the iitial value problem y (5.) y 4, y 6 Solutio. We observe that all poits ecept are ordiary poits for the differetial equatio (5.). Sice the iitial values of y ad its first derivative are prescribed at =, we assume Istitute of Lifelog Learig, Uiversity of Delhi Page 5

16 Power series solutio of Differetial equatios about ordiary poits y( ) as the geeral solutio of (5.). Differetiatig term by term, we obtai, Substitutig the values of y,, ad i (5.), we have. To make the epoet of same i each term, we replace by i secod sum ad by i fourth sum we get,. The commo rage of these terms is form to. We write the terms for = ad = i each sum separately ad thus above equatio takes the form 6. ombiig like powers of, this takes the form ( 6 ) usig idetity priciple, we get =, 6 ad 6 The recurrece formula gives,. Istitute of Lifelog Learig, Uiversity of Delhi Page 6

17 Power series solutio of Differetial equatios about ordiary poits 8 Put =, 4 Put =, Substitutig the values of,, 4, 5 ito the assumed solutio of (5.), we get OR 4 5 y( ) y( ) This represets the geeral solutios of the differetial equatio (5.). y 4 4, y 6 6. Hece the particular solutio is give by 5 4 y( ) y( ) Eample 5. Solve the iitial value problem t t t y dt dt (5.) y 4 ad y Solutio: Sice the iitial coditios are give at t =, we will assume a power of (5.). series solutio of the form y t t Also, we ote that t = is a ordiary poit of (5.). Here, we will ot substitute the assumed series solutio i (5.) as doe i other problems, istead we first substitute = (t-) i (5.), so that we ed up fidig a series of the form. Now the substitutio ad t chages t t 4 to Istitute of Lifelog Learig, Uiversity of Delhi Page 7

18 Power series solutio of Differetial equatios about ordiary poits. dt dt d y d d y dt dt. Hece, equatio (5.) chages to 4 y correspodig to t =. with iitial coditios y 4 & y This is the iitial value problem we solved i Eample of this sectio. The particular solutio obtaied i Eample was y( ) Replacig by t, we have the required particular solutio 4 y t 4 t t t t... (5.) 6 Also, ote that the solutio i terms of coverges for Hece (5.) coverges for t or for t. Eercise Fid power series solutio i powers of of differetial equatios from 6. Also, fid the radius of covergece i each case.. y. 4 y y y y y Fid power series solutio of the iitial value problems i 7. Istitute of Lifelog Learig, Uiversity of Delhi Page 8

19 Power series solutio of Differetial equatios about ordiary poits 7. y y y. y ; y, y ; ; y, y ; y, y y, y Solve the iitial value problems i 4. Also, fid the iterval of covergece. y.. ; d y 6 4y. y, y d y 6 4 6y d y 6 9 y The Legedre s Equatio The secod order liear differetial equatio ; y, y ; y, y ; y, y y (6.) is called the Legedre s equatio of order, where the real o. satisfies the iequality >. A solutio to this equatio is called a Legedre fuctio. This differetial equatio has wide rage of applicatios. Value additio: Note Istitute of Lifelog Learig, Uiversity of Delhi Page 9

20 Power series solutio of Differetial equatios about ordiary poits Adrie Marie Legedre iveted Legedre Polyomials (the cotributio for which he is best remembered) i the cotet of gravitatioal attractio of ellipsoids. Legedre was a fie Frech Mathematicia who suffered the misfortue of seeig most of his best work i elliptic itegrals, umber theory ad the method of least squares- superseded by the achievemets of youger ad abler me. For istace, he devoted 4 years to the stu of elliptic itegrals, ad his two volume treatise o the subject had scarcely appeared i prit before the discoveries of Abel ad Jacobi revolutioized the field. Legedre was remarkable for the geerous spirit with which he repeatedly welcomed ewer ad better work that made his ow obsolete. Writig the Legedre equatio i ormalized form, we have d y y. learly, = is a ordiary poit ad =,- are its oly sigular poits. The Legedre equatio has two liearly idepedet solutios epressible as power series i with radius of covergece at least. Let y( ) be the power series solutio of (6.) Differetiatig term by term, we get ad. Substitutig these values i (6.), we have To make the epoet of same i all summatios, we replace by + i first summatio. Also we shift the ide of sum from = to = i secod term ad from = to = i third term. This gives. By idetity priciple. Thus, Istitute of Lifelog Learig, Uiversity of Delhi Page

21 Power series solutio of Differetial equatios about ordiary poits, Put =,,,, we have,!!, 4 4! 4 5 5! Geeralizig the above patter, we have Let......! ! a......! a !. The, a, ad a,. We have two power series solutios of Legedre's equatio of order which are liearly idepedet ad y a y a Ad the geeral solutio is liear combiatio of y ( ) ad y ( ). Now, suppose m, a o-egative iteger. If m is eve, the a whe > m. I this case, y is a polyomial of degree m ad y (o-termiatig). If m is a odd iteger, the a whe + > m. is a ifiite series Istitute of Lifelog Learig, Uiversity of Delhi Page

22 Power series solutio of Differetial equatios about ordiary poits I this case, y is a polyomial of degree m ad y ( ) ifiite series. is a o-termiatig Thus i either case, oe of the two solutios is a polyomial or other is a otermiatig series. With a appropriate choice of the arbitrary costats (m eve) ad (m odd), the Legedre s equatio has polyomial solutio. The m th degree polyomial solutio of Legedre s equatio of order m, ( ) y '' y ' m( m ) y, is deoted by P ( ) ad is called the Legedre polyomial of degree m. It is m customary to choose a arbitrary costat so that the coefficiet of (!) / [ (!) ]. It the turs out that i P ( ) is ( ) ( k)! N k k ( ), k k!( k)!( k)! P where N /, the itegral part of /. The first si Legedre polyomials are P ( ), P ( ), P ( ) ( ), P ( ) (5 ), 4 5 P4 ( ) (5 ), P5 ( ) (6 7 5 ). 8 8 Graphs y P ( ) of the Legedre polyomial for,,,,4,5. Istitute of Lifelog Learig, Uiversity of Delhi Page

23 Power series solutio of Differetial equatios about ordiary poits Summary I this lesso we have defied ad emphasized o the followig: Secod order homogeous liear differetial equatio Power Series Radius of covergece Ordiary poits, sigular poits Solutio about a ordiary poit by power series method Legedre's equatio Refereces [].H. Edwards ad D.E. Pey, Differetial Equatios ad boudary Value Problems: omputig ad Modelig, Pearso Educatio, Idia, 5. [] S. L. Ross, Differetial equatios, rd editio, Joh Wiley ad Sos, Idia, 4. [] William R. Derrick ad Staley L. Grossma, A First ourse i Differetial Equatios, Third Editio. Istitute of Lifelog Learig, Uiversity of Delhi Page