Subject: Differential Equations & Mathematical Modeling-III

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1 Power Series Solutios of Differetial Equatios about Sigular poits Subject: Differetial Equatios & Mathematical Modelig-III Lesso: Power series solutios of differetial equatios about Sigular poits Lesso Developer: Nisha Bohra ollege/departmet: Sri Vekateswara ollege, Uiversity of Delhi Istitute of Lifelog Learig, Uiversity of Delhi

2 Power Series Solutios of Differetial Equatios about Sigular poits Table of otets. Learig Outcomes. Itroductio 3. lassificatio of Sigular Poits Eercise 4. Solutio about Regular Sigular Poits 4.. The Method of Frobeius 4... Frobeius series solutio whe r r is ot a iteger 4.. Frobeius series solutio whe r r is a iteger Eercise 5. Bessel s Equatio 5.. Bessel s equatio of order zero 5.. Gamma fuctio 5.3. Bessel s equatio of order p, p> Summary Refereces Istitute of Lifelog Learig, Uiversity of Delhi P a g e

3 Power Series Solutios of Differetial Equatios about Sigular poits. Learig Outcomes After readig this lesso reader will be able to uderstad the followig Sigular poits Types of sigular poits Frobeius Method Gamma fuctio Bessel s equatio ad Bessel fuctio. Itroductio Let us start by recallig the defiitios of ordiary poit ad sigular poit of a differetial equatio. osider the secod order homogeeous liear differetial equatio A B y d d (.) where the coefficiets A, B ad are aalytic fuctios of. Rewrite equatio (.) as P Q y d d with P B / A ad / Q A. (.) Equatio (.) is called equivalet ormalized form of equatio (.). Note that P ad Q are ot aalytic at poits where A is zero. Defiitio. A poit = a is called a ordiary poit of the differetial equatio (.) ad the equivalet equatio (.) 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. Now, cosider the homogeeous secod order liear differetial equatio A B y d d (.3) where the coefficiets A, B ad are cotiuous real fuctios of o a iterval I ad A() is ot idetically zero o I. If =a is a ordiary poit of the differetial equatio (.3), the equatio (.3) has two liearly idepedet solutios, each of the form y a (.4) Istitute of Lifelog Learig, Uiversity of Delhi 3 P a g e

4 Power Series Solutios of Differetial Equatios about Sigular poits ad these power series solutios coverges i some iterval a R about a (R > ). However, if = a is a sigular poit of (.3), the we are ot assured of a power series solutio of (.3) i powers of (-a). Ideed a equatio of the form (.3) with a sigular poit at = a does ot, i geeral, have a solutio of the form (.4). learly we must seek a differet type of solutio i such a case. It happes that uder certai coditios we are justified i assumig a solutio of the form r y a a, where r is a certai (real or comple) costat. I order to state coditios uder which a solutio of above form is assured, we proceed to classify sigular poits 3. lassificatio of Sigular Poits osider the homogeeous secod order liear differetial equatio A B y. (3.) d d We first write the differetial equatio (3.) i the form P Q y, (3.) d d where P B / A ad Q / A. Defiitio. Let = a be a sigular poit of equatio (3.), i.e., at least oe of the fuctio a P ad from P ad Q i equatio (3.) is ot aalytic at = a. If the fuctios a Q are both aalytic at = a, the = a is called a regular sigular poit of the differetial equatio (3.). If either or both the fuctios a P ad a Q are ot aalytic at = a, the it is called a irregular sigular poit. Eample. lassify the sigular poits of the give differetial equatio 5 y d d. (3.3) Writig this is ormalized form, we have d P 5 y d 5, Q Sice both P & Q are ot aalytic at =, we coclude that = is a sigular poit of (3.3). Istitute of Lifelog Learig, Uiversity of Delhi 4 P a g e

5 Power Series Solutios of Differetial Equatios about Sigular poits Now, P ad Q 5 are both aalytic at =. So, = is a regular sigular poit of differetial equatio (3.3). Eample. lassify the sigular poits of the give differetial equatio d y dy y d d. (3.4) I ormalized form, equatio (3.4) is d y dy y. d d Here, P, Q learly = ad = are sigular poits of equatio (3.4). osider = first, P, Q Q is aalytic at =, but P is ot aalytic at =. Hece = is a irregular sigular poit of (3.4). Now, cosider = P, Q Both are aalytic at =, ad hece = is a regular sigular poit of differetial equatio (3.4). Eample 3. lassify the sigular poits of the give differetial equatio 4 d y dy si cos y. (3.5) d d Writig equatio (3.5) i ormalized form d y si dy cos y 4 d d si Here, P, Q 4 cos Istitute of Lifelog Learig, Uiversity of Delhi 5 P a g e

6 Power Series Solutios of Differetial Equatios about Sigular poits Writig P ad Q ! 5! ! 5! ! 4! 6! ! 4! 6!...! 4! 6! Observe that, = is a sigular poit of (3.5). Now, cosider ad Q 3 P... 3! 5! ! 5!...! 4! 6! 4...! 4! 6! Oe ca ote that P ad Q are aalytic at =. Thus, = is a regular sigular poit of (3.5). Remark. We will discuss the problems where = is a sigular poit of equatio (3.), for if = a is a sigular poit of a differetial equatio, the it ca be easily be trasformed ito oe havig a correspodig sigular poit at = by the substitutio t a. For eample, cosider the differetial equatio y. d d I ormalized form, it is Istitute of Lifelog Learig, Uiversity of Delhi 6 P a g e

7 Power Series Solutios of Differetial Equatios about Sigular poits d y dy y d d learly = is a sigular poit. Put t = = t +. The dy dy dt dy d dt d dt d y d y dt d y d dt d dt So, the equatio chages to t t t y dt dt The ew equatio has sigular poit at t =. Eercise heck whether = is a ordiary poit or sigular poit. If it is a sigular poit, classify it. cos y d d y d d 3y d d 3. d y dy si y d d 3 4. e y d d Solutio about Regular Sigular poits We shall state a basic theorem (without proof) cocerig the eistece of solutio about a regular sigular poit. Theorem. Let = a be a regular sigular poit of the differetial equatio (3.). The the differetial equatio has at least oe o-trivial solutio of the form r y a a, where r is a costat (real/comple) which may be determied ad this solutio is applicable i some deleted iterval a R about a. Istitute of Lifelog Learig, Uiversity of Delhi 7 P a g e

8 Power Series Solutios of Differetial Equatios about Sigular poits r Note that at least oe solutio of the form y a a about the regular sigular poit = a of differetial equatio (3.) is guarateed by above theorem. The atural questio that arises is, What is the procedure to determie the coefficiets ad umber r i this solutio? I the followig sectio, we discuss the procedure to obtai the values of ad r. 4.. The Method of Frobeius The method of Frobeius is amed after Ferdiad Georg Frobeius. It is used to fid a ifiite series solutio of a secod order homogeous liear differetial equatio about a regular sigular poit. We shall cosider the case whe = is a regular sigular poit of the differetial equatio. I the method outlied here, we shall seek solutios valid i some iterval R. To obtai solutio valid for R, simply replace by ad proceed as i the outlie. Let us first see the workig of this method. osider the differetial equatio (3.) A B y d d ad its equivalet ormalized form (3.) P Q y d d where P B / A ad Q / A. Let = be a regular sigular poit of (3.) P ad Q are aalytic at =. Let p P ad q Q The (3.) chages to. dy q d y p y d d (4.) p q y. (4.) d d or Sice p ad q are aalytic fuctios at =, so 3 p p p p p q q q q q3... (4.3) Istitute of Lifelog Learig, Uiversity of Delhi 8 P a g e

9 Power Series Solutios of Differetial Equatios about Sigular poits Let r r (4.4) y be solutio of (4.), where. (4.4) is called the Frobeius series. Term wise differetiatio i (4.4) leads to dy d r r ad dy d r r r. Substitutig the values of dy d y y, & i (4.) ow yields d d r r r r r r r... q q r r P P... r r... r The lowest power of that appears i above equatio is r. If the above equatio is to be satisfied, the the coefficiet r r must vaish. Because, r must satisfy pr q of r r r p r q. (4.5) Equatio (4.5) is called idicial equatio of (4.) ad its two roots are the epoets of the differetial equatio at the poit =. Let r ad r be roots of the idicial equatio. The r must be oe of r ad r. ase I. If r r Two Frobeius series solutio are possible. ase II. If r r There is oly oe possible Frobeius series solutio. After fidig the epoets r ad r, we fid coefficiets i Frobeius series solutio by substitutig the series (4.4) ad values of dy d ad dy i the differetial equatio. d Remark. We will cofie our discussio to the case whe r ad r, both are real. Istitute of Lifelog Learig, Uiversity of Delhi 9 P a g e

10 Power Series Solutios of Differetial Equatios about Sigular poits Eample. Fid the epoets ad the possible Frobeius series solutios of the equatio 3 y 3 y y. Solutio. Rewrite the equatio as i (4.) 3 y y y. 3. p, q 3 Thus p, q So, idicial equatio is 3 r r r. r r r r r, r. The the two possible Frobeius series solutio are of the form y a / ad Value Additio: Note y b If p ad q are polyomials i, the p p ad p ad q are idetermiate forms at =, the p p lim p lim P ad q q lim q lim Q. q q. But if Now, we state the followig theorem without proof. Theorem. Suppose = is a regular sigular poit of the differetial equatio (4.) y '' p y q y. Let R > deote the miimum of the radii of covergece of the power series p q. p & q Let r ad r be the roots with r r The r r p r q. of the idicial equatio Istitute of Lifelog Learig, Uiversity of Delhi P a g e

11 Power Series Solutios of Differetial Equatios about Sigular poits a) For >, a solutio of the form r y a a correspodig to the larger root r. b) If r r is either zero or a positive iteger, the a secod liearly idepedet solutio for > of the form r y b b correspodig to the smaller root r. The radii of covergece of the power series give i a) ad b) are each at least R. The coefficiets i these series ca be determied by substitutig the series (4.4) i the differetial equatio y '' p y q y. 4.. Frobeius series solutio whe r r is ot a iteger Eample. Fid the Frobeius series solutio of 3 y d d (4.6) Solutio. First we write the equatio as i (4.) 3 d y dy y. d d 3 d y dy y. d d 3 p, q Hece. p p, q q p ad 3 q are polyomials, so the Frobeius series solutio will coverge for > The idicial equatio is give by r r p r q. 3 r r r. Istitute of Lifelog Learig, Uiversity of Delhi P a g e

12 Power Series Solutios of Differetial Equatios about Sigular poits. r r So the epoets are r ad r. Sice r r is ot a iteger, so by Theorem we have two liearly idepedet Frobeius series solutio / y ( ) a ad y( ) b Istead of substitutig y ad y separately, we begi by substitutig r y( ) (4.7) i (4.6). We will obtai a recurrece relatio depedig o r. Differetiatig (4.7) term by term, we get dy d r r ad dy d r r r. Substitutig the values of dy d y y,, i (4.6), we get d d r 3 r r r r r r (4.8) To make the epoet of same i all the summatios, we replace by i the third summatio. This gives r r r 3 r r r r. The commo rage of summatio is from to, so we write the terms for = ad = from each sum separately ad we cotiue to use the sigma otatio to deote the remaider of sum. This gives r r r r 3r r r 3 r r r r 3 r. Idetity priciple yields Istitute of Lifelog Learig, Uiversity of Delhi P a g e

13 Power Series Solutios of Differetial Equatios about Sigular poits r r r 3 OR r r r 3 r r. (4.9) OR r 5r. (4.) r r 3 r. (4.) From (4.9), we have r r which represets the idicial equatio ad its solutios are: r or r. From (4.) we have as r 5r Usig (4.), we have the followig recurrece relatio r r for (4.) ase I r Now, we write a i place of ad substitute r i (4.). a a 3, puttig =, 3, 4, 5, 6, we have a 4 a, 3 a as a a a a4, 5 a. Ad a4 a a So, the first Frobeius solutio is 4 6 y a ase II r = Istitute of Lifelog Learig, Uiversity of Delhi 3 P a g e

14 Power Series Solutios of Differetial Equatios about Sigular poits Write b i place of ad substitute r = i equatio (4.), we have b b 3, Puttig =, 3, 4, 5, 6, we get b b, 3 b, b b 4 b4, b5 ad b b b6. Hece the secod Frobeius solutio is 4 6 y b Eample 3. Fid the Frobeius series solutio of 5 y d d (4.3) Solutio. We write the differetial equatio (4.3) i the form (4.) 5 d y dy y d d p ad q p 5 p, q q 5. Sice p ad q are polyomials, the Frobeius series will coverge for >. The idicial equatio is r r p r q 5 r r r r 3r 5 5 r ad r are the epoets of give differetial equatio. Sice r r is ot a iteger, Theorem guaratees the eistece of two liearly idepedet Frobeius series solutio Istitute of Lifelog Learig, Uiversity of Delhi 4 P a g e

15 Power Series Solutios of Differetial Equatios about Sigular poits 5 / y a ad y b As doe i previous eample, we assume r y( ) as the solutio of (4.3). dy d The r r dy d r r r Substitutig the values of y, dy & d dy d i (4.3) we get r r r r r r r 5. To make the epoet of same, we replace by i the third summatio r r r r r r r 5. Sice the commo rage is from to, simplifyig as i previous Eample, we get r r r r r 5 [(( r)( r ) ( r) 5) ]. Equatig to zero the coefficiet of r r r 5 Or r, we have r 3r 5. Sice, r 3r 5 (the idicial equatio) Equatig to zero the coefficiet of r, we have r r r 5 5 r r r,. (4.4) 5 ase I. r We write a i place of ad substitute 5 r i (4.4) Istitute of Lifelog Learig, Uiversity of Delhi 5 P a g e

16 Power Series Solutios of Differetial Equatios about Sigular poits a a 7, Puttig =,, 3 we get a, a a, a a a a a Thus the first Frobeius solutio is 5/ 3 y a ase II r Now, we write b i place of ad substitute r = i (4.4). b b 7, Put =,, 3, we get b b, b b b, b b b Hece the secod Frobeius solutio is y b Frobeius series solutio whe r r is a iteger I case, r r is a positive iteger, the Theorem guaratees oly the eistece of the Frobeius series solutio correspodig to the larger epoet r. I the followig eample, we will observe that a secod Frobeius series solutio is obtaied usig the series method. Eample 4. Fid the Frobeius series solutio of y d d (4.5) Solutio. We first write the equatio i form (4.) d y d dy y d d y dy or y. d d The = is a regular sigular poit ad p, q. The idicial equatio is Istitute of Lifelog Learig, Uiversity of Delhi 6 P a g e

17 Power Series Solutios of Differetial Equatios about Sigular poits r r r rr The epoets are r = ad r =. I this case r r. So, we caot proceed as i the previous two eamples. I this case, we begi with the smaller epoet. The eistece of secod Frobeius solutio will the deped o the recurrece relatio. Substitutig y = i (4.5), we get.. Shiftig the ide by i secod sum, we obtai. The case = ad = reduces to. = ad. = Thus we have two arbitrary costats ad, therefore we may epect a geeral solutio ivolvig two liearly idepedet Frobeius series solutios. We have the followig recurrece relatio, Puttig =, 3, 4, 5, 6, 7 gives, , 4.3 4! 6.5 6! 6 4, 5.4 5! ! 7 5 I geeral, Istitute of Lifelog Learig, Uiversity of Delhi 7 P a g e

18 Power Series Solutios of Differetial Equatios about Sigular poits! ad!, Hece, geeral solutio of equatio (4.5) is give by y ! 4! 3! 5!!! y cos si Or y y y where y ad y cos si are Frobeius series solutios of (4.5). Value Additio :NOTE If the solutio eists for smaller epoet, as i the above eample, the the give differetial equatio will always have two liearly idepedet solutios. However, if the solutio does't eist for smaller epoet we will repeat the process with the larger epoet to obtai the Forbeius solutio guarateed by Theorem. Eample 5. Fid the Frobeius series solutio of d y dy y d d (4.6) Solutio: We first write the equatio as d y dy ( ) y d d q -. Hece = is a regular sigular poit with p ad So, the solutio will coverge for >. p, q Istitute of Lifelog Learig, Uiversity of Delhi 8 P a g e

19 Power Series Solutios of Differetial Equatios about Sigular poits Hece the idicial equatio is r r p r q r r. Thus, the epoets are r = ad r = -. I this case r r = (a iteger). As doe i previous eample, we first start with smaller epoet ( ). Substitutig y = i (4.6), we get 3 ( ) ( ) ( )( ). Or. Replacig by (-), i third term, we get. This implies [ ], which implies = ad Thus, we have the followig recurrece relatio ( ) +,, for. which for =, yields,. which implies =, a cotradictio. Hece Frobeius series solutio does ot eist for smaller epoet. But by Theorem, solutio does eists for larger epoet. So, ow we begi with larger epoet. Substitutig, y = i (4.6), we get ( ) ( ) ( )( ). 3 Or. Istitute of Lifelog Learig, Uiversity of Delhi 9 P a g e

20 Power Series Solutios of Differetial Equatios about Sigular poits Replacig by (-) i third term, we get. This gives [( ) ] 3, for. which implies = ad =, ( ) Puttig =, 3, 4, 5, 6, 7, we get =, 3,.4!.!. 4 =, 5, !.3!. 4 6 =, !.4!. Geeralizig the above patter, we obtai for odd Ad ( )!( )!. Hece, the Frobeius series solutio correspodig to larger epoet is y( ) ( )!( )!. Eercise Use the method of Frobeius to fid solutios ear = for each of the differetial equatio.. 4 y d d y d d. Istitute of Lifelog Learig, Uiversity of Delhi P a g e

21 Power Series Solutios of Differetial Equatios about Sigular poits 3 y d d y d d d y dy 4y d d 5. Use the method of eample 4 ad 5 to fid two liearly idepedet Frobeius series solutios of the differetial equatio i followig problems.. 9y d d. 4 8 y d d y d d 5. Bessel s Equatio The differetial equatio d y dy p y, p is a parameter, d d is called Bessel's equatio of order p amed after Germa Astroomer Friedrich Wilhelm Bessel. A solutio of Bessel's equatio of order p is called a Bessel fuctio of order p. Bessel's equatio ad Bessel fuctio occur i coectio with may problems of physics ad egieerig. Value Additio Friedrich Wilhelm Bessel ( ) was a distiguished Germa Astroomer. Bessel was the first ma to determie accurately the distace of a fied star (the star 6 ygi). I 844, he discovered the biary (or twi) star Sirius. The compaio star to Sirius has the size of a plaet but the mass of a star, its desity is may thousads of times the desity of water. It was the first dead star to be discovered, ad it occupies a special place i the moder theory of stellar evolutio. 5.. Bessel's equatio of order zero osider y d d I the ormalized form (5.) becomes d y dy y. d d. (5.) Istitute of Lifelog Learig, Uiversity of Delhi P a g e

22 Power Series Solutios of Differetial Equatios about Sigular poits Hece = is a regular sigular poit with p ad q So, the solutio will coverge for >. Here p ad q. Hece the idicial equatio is r r p r q r r r r. Thus we have oly oe epoet r = ad so there is oly oe Frobeius series solutio y of (5.) guarateed by Theorem. Substitutig y i (5.), we have.. We shift the ide of summatio from = to = i the first term ad from = to = i the secod term. The, we combie the first two terms ad shift the ide of sum i the third term by to obtai.. ad, for odd. Substitutig =, 4 ad 6 i the recurrece relatio, we get, Geeralizig the above patter, Istitute of Lifelog Learig, Uiversity of Delhi P a g e

23 Power Series Solutios of Differetial Equatios about Sigular poits.4...! The solutio is y! The choice = gives us the Bessel fuctio of order zero of the first kid, deoted by J (), oe of the most importat special fuctio i mathematics. So J!. Before we study the Bessel s equatio of order p, we will have a brief itroductio about Gamma fuctios. 5.. Gamma fuctio The gamma fuctio ( ) is defied for > by t e t dt. ( ) This improper itegral coverges for each >. (Studets must have verified it i their Improper itegrals classes). I fact, gamma fuctio is a geeralizatio (for > ) of the factorial!, which is defied oly if is oegative iteger. To verify that ( ) is a geeralizatio of!, we first ote that t t b e dt e b (5.) () lim. The we itegrate by parts takig u t ad dv t e dt b b b b t t t t ( ) lim e t dt lim e t e t dt lim e t dt b b b that is, ( ) ( ) (5.3) It is a very importat property of the gamma fuctio. Istitute of Lifelog Learig, Uiversity of Delhi 3 P a g e

24 Power Series Solutios of Differetial Equatios about Sigular poits ombiig equatios (5.) ad (5.3), we have (). ()!, (3). ()!, (4) 3. (3) 3!, I geeral, ( )! for. Aother sigificat special value of the gamma fuctio is t / u e t dt e du, where we have substituted u for t i the first itegral. The fact that e u du is kow, but is far from obvious. Although ( ) is defied by meas of a itegral oly for >, we ca also use the recursio formula i (5.3) to defie ( ) wheever is either zero or a egative iteger. Thus ( ) is defied for all,,, Bessel s equatio of order p, p> osider the differetial equatio ( p ) y d d (5.4) I the ormalized form (5.4) becomes d y dy ( p ) y. d d learly, = is a regular sigular poit of (5.4) with p ad So, the solutio will coverge for >. Here, p ad q p. Hece the idicial equatio is r r p r q r r r p r p. q ( p ). Istitute of Lifelog Learig, Uiversity of Delhi 4 P a g e

25 Power Series Solutios of Differetial Equatios about Sigular poits Thus, the epoets are r = p ad r = -p ad r r p. ase : r r p is ot a iteger. By Theorem, the differetial equatio (5.4) will have two liearly idepedet Frobeius series solutios p y a ad p y b ase : r r p is a iteger. Agai by Theorem, we are oly certai of a Frobeius series solutio correspodig to the larger root r = p. We shall ow proceed to obtai this oe solutio first, eistece of which is always assured. We begi by substitutig r y (5.5) i (5.4). Differetiatig (5.5) term by term, we get dy d r r ad dy d r r r. Substitutig the values of dy d y y,, i (5.4), we have d d r r r ( r p ) [( r ) p ] [(( r) p ) ]. Equatig to zero each power of ad usig the fact, we have r p (5.6) (( r ) p ) (5.7) (( r) p ), (5.8) Equatio (5.8) represets the idicial equatio of the differetial equatio (5.4). Its roots are p. ase. r p Now lettig r r p ad writig a i place of, (5.7) gives a. Istitute of Lifelog Learig, Uiversity of Delhi 5 P a g e

26 Power Series Solutios of Differetial Equatios about Sigular poits From (5.8), we obtai the followig recurrece formula a a ( ), p (5.9) Sice a, a for odd. Puttig =, 4,6 we get a a a a, a, ( p) 4(4 p).4( p)(4 p) 4 a 6 a a 4 6(6 p).4.6( p)(4 p)(6 p) Thus we have a for odd ad a ( ) a ( )[( p)(4 p)...( p)] ( ) a =![( p)( p)...( p )],. Hece the solutio of the differetial equatio (5.4) correspodig to the larger root p is give by y ( ) a ( ) p![( p)( p)...( p)]. (5.) If p is a positive iteger, we may write y ( ) as p ( ) y( ) a p!!( p)! p If p is ot a positive iteger, we eed a geeralizatio of the factorial otatio which is provided by gamma fuctio. Hece (5.) takes the form p ( ) y( ) a ( p )! ( p ) p. (5.) Thus the solutio of differetial equatio (5.4) correspodig to the larger root is give by (5.). p If we set the arbitrary costat a i (5.) as multiplicative iverse of ( p ), we obtai a particular solutio of (5.4). This particular solutio is very importat fuctio deoted by fuctio of first kid of order p. That is, J p is defied by J p ad is called the Bessel Istitute of Lifelog Learig, Uiversity of Delhi 6 P a g e

27 Power Series Solutios of Differetial Equatios about Sigular poits ( ) Jp( )! ( p ) p. ase. r p Usig r = -p ad b i place of, equatios (5.7) ad (5.8) gives ( p ) (5.) ( p) b b (5.3) We see that if p is a positive iteger i.e., if p is either a positive iteger or a odd positive itegral multiple of ½ the there is a difficulty. For the, whe, p equatio (5.3) reduces to. b b. Thus, if b, the o value of b ca satisfy this equatio. But if p is a odd positive itegral multiple of ½, we ca overcome this difficulty. We eed oly choose b for all odd values of. Hece if p is ot a (positive) iteger, we take b for odd ad defie the coefficiets of eve subscript i terms of b by meas of the recursio formula b b p, (5.4) omparig (5.4) with (5.9), we see that (5.4) will lead to the same result as that i (5.), ecept with p replaced with p. Thus the secod solutio is give by y ( ) b ( ) p![( p)( p)...( p)]. y ( ) ca be writte i terms of Gamma fuctio as p p ( ) ( ) ( )! ( p ) y b p. Takig b as p / [ ( p )], we obtai the secod liearly idepedet solutio J p ( ) ( )! ( p ) p. Hece, if p is ot a iteger, we have the geeral solutio y( ) c J ( ) c J ( ) p p of the Bessel s equatio of order p for >. Istitute of Lifelog Learig, Uiversity of Delhi 7 P a g e

28 Power Series Solutios of Differetial Equatios about Sigular poits The followig graph suggest that J ad J behavior ad that the positive zeroes of J ad J separate each other. each have a damped oscillatory This is ideed the case. I fact it may be show that for each p, the Bessel s fuctio J p ( ) has a damped oscillatory behavior as ad the positive zeroes of J p ad J p separate each other. Remark: While fidig the solutio correspodig to smaller root, we have ot discussed the case, whe p is a positive iteger. The solutio obtaied i this case is p p p p ( p )! Yp( ) l Jp( ) ( )! k k k k!( p)! where is a Euler costat. The solutio Y p is called the Bessel fuctio of the secod kid of order p. Thus if p is a positive iteger, two liearly idepedet solutios of the differetial equatio (5.) are J ad Y. Also, if p is a positive iteger, the geeral solutio of Bessel s equatio of order p is give by y( ) c J ( ) c Y ( ) p p p p where c ad c are arbitrary costats. For detailed procedure of obtaiig this secod solutio Y p, reader ca look at the refereces metioed at the ed of the chapter. Summary I this lesso we have defied ad emphasized o the followig: Sigular poits Types of sigular poits Frobeius Method Gamma fuctio Bessel s equatio ad Bessel fuctio Istitute of Lifelog Learig, Uiversity of Delhi 8 P a g e

29 Power Series Solutios of Differetial Equatios about Sigular poits 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, 3 rd editio, Joh Wiley ad Sos, Idia, 4. [3] William R. Derrick ad Staley L. Grossma, A First ourse i Differetial Equatios, Third Editio. Istitute of Lifelog Learig, Uiversity of Delhi 9 P a g e

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +

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