After the completion of this section the student. V.4.2. Power Series Solution. V.4.3. The Method of Frobenius. V.4.4. Taylor Series Solution

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Chapter V ODE V.4 Power Series Solutio Otober, 8 385 V.

1 Chapter V ODE V.4 Power Series Solutio Otober, V.4 Power Series Solutio Objetives: After the ompletio of this setio the studet - should reall the power series solutio of a liear ODE with variable oeffiiets - should reall the defiitios of ordiary ad sigular poits - should reall the Frobeius method of the power series solutio about a regular sigular poit Cotets: V.4.. Defiitios V.4.. Power Series Solutio V.4.3. The Method of Frobeius V.4.4. Taylor Series Solutio V.4.5. Power Series Solutio with Maple V.4.6. Review Questios ad Eerises

2 386 Chapter V ODE V.4 Power Series Solutio Otober, POWER SERIES SOLUTION V.4.. DEFINITIONS I the previous setio we studied how to solve a liear ODE with ostat oeffiiets, ad we disovered that for them we always a fid liearly idepedet solutios (fudametal set). For oe speial form of a liear ODE with variable oeffiiets, amely the Euler-Cauhy equatio, we a fid the solutio by redutio to a equatio with ostat oeffiiets. I geeral, for liear equatios with variable oeffiiets, the fudametal set aot be determied by some uiversal azats their solutios have a wide variety of futioal forms. But if the oeffiiets of the differetial equatio are well eough behaved futios, the the solutios of the liear ODE are aalytial futios (reall Setio II.4) ad they a be represeted by the power series epasio (aalyti futio) about some poit : () + ( ) + + ( ) + ( ) y where oeffiiets have to be determied. y power series power series about y power series about overgee Series overges at if the sequee of partial sums overges N ( ) S S overges. absolute overgee if the series of absolute values radius of overgee R lim + R lim iterval of overgee diverges overges diverges R + R absolutely overges for all R < < + R Covergee at boudary poits ± R has to be ivestigated separately. aalyti futio Futio is alled aalyti at if it has a Taylor series epasio about. Power series defies a aalyti futio i its iterval of overgee: f, ( R, R + ) shift of ide + m a am m + m + m dummy ide idetity theorem If for all, the all oeffiiets

3 Chapter V ODE V.4 Power Series Solutio Otober, operatios with series Let f ( ) a ad overge i ( R, R + ) : summatio f ( ) ± g( ) ( a ) ± b g b multipliatio f ( ) g( ) a b( ), ab differetiatio f a f a itegratio f ( ) d a( ) d ( ) a + + Taylor series f ( ) ( ) ( ) ( ) f! ( ) f f f ( ) !! Malauri series f ( ) ( f )! ( ) f f !! f Table of Taylor series si 5. os 6. sih 7. osh < < < < ( ) < < ( ) ( + ) + <! ( ) <! ( + ) < < + < <! < <! 8. l( + ) ( ) + < < + 9. e < <!

4 388 Chapter V ODE V.4 Power Series Solutio Otober, 8 d order ODE I our aalysis, without sigifiat loss of geerality, we restrit ourselves to solutio of the d order homogeeous liear ODE for D : a y + a y + a y () Dividig equatio () by the leadig oeffiiet a ormal form we a rewrite it i the y + p y + q y (3) a ( ) p( ) ad q( ) where oeffiiets are p( ) a ( ) ad a ( ) q a ( ). However, we do ot require a ( ) for all D. We will distiguish the poits of the domai by the followig riteria: ordiary poit The poit a. is alled the ordiary poit of the ODE (), if sigular poit The poit a. is alled the sigular poit of the ODE (), if regular sigular poit The sigular poit is alled the regular sigular poit of ODE (3), if + ( ) + ( ) + ( ) p p p p... p (4) q( ) q + q( ) + q( ) +... q ( ) are aalyti at. (5) Eamples ) If the poit is a ordiary poit the we will be able to fid two liearly idepedet solutios of ODE () i the form of the power series () about the ordiary poit. If the poit is a sigular poit, the for the speial ase of the regular sigular poit we will be able to fid oe solutio i the form of the power series epasio about the sigular poit, ad the seod solutio a be ostruted i some speial way by the Frobeius method. y + y + si y all poits are ordiary y + y + y is the oly sigular poit ) 3) y y y i, i are the sigular poits y + y y 4) y + y y p ( ) + is aalyti, p q is aalyti, q Therefore, is a regular sigular poit.

5 Chapter V ODE V.4 Power Series Solutio Otober, V.4.. POWER SERIES SOLUTION R R iterval of overgee sigular poit + R Theorem (power series solutio about the ordiary poit) Let all oeffiiets a of ODE () be aalyti i some ope iterval D, ad let D be a ordiary poit of ODE (). The two liearly idepedet solutios of ODE () a be foud i the form of the power series y( ) () with the radius of overgee at least (a be bigger) R where is the earest to a sigular poit. I other words, if oeffiiets of () are aalytial futios, the aalytial solutio of () a be foud. The proedure of fidig the solutio briefly will osist of the followig steps:. Choose a ordiary poit (usually,, if ot, the by the hage of variable ξ, the ordiary poit a be traslated to ). y. Assume the solutio i the form ad substitute it ito ODE () (reall that overget power series a be differetiated term by term). 3. Combie the obtaied equatio ito a sigular power series ad use the Idetity Theorem to establish the reurree relatio for oeffiiets. Appliatio of the reurree relatio will yield two sets of oeffiiets otaiig the arbitrary ostats ad. 4. Eah set of oeffiiets will produe a liearly idepedet solutio i the form of the power series (). The both solutios a be tested for the radius of overgee. Eample (power series solutio about the ordiary poit) Solutio: Fid the solutio of the differetial equatio y + y + y (6) We will loo for the solutio of this liear ODE with variable oeffiiets i the form of power series aordig to Theorem. All oeffiiets of the give equatio are aalyti for all. There are o sigular poits of this equatio.. Choose a ordiary poit. Assume the solutio i the form y. Fid the derivatives of the assumed solutio (differetiate term by term): y (summatio effetively starts with ) (summatio effetively starts with y Substitute ito equatio (5): )

6 39 Chapter V ODE V.4 Power Series Solutio Otober, 8 3. To ombie all sigma-terms i a sigle summatio we eed all of them to have the same ide i the powers of m ad the same startig ide of summatio m m. First, reame the idies i eah sigma-term (reall that ide of summatio is dummy ad that the same result a be ahieved with ay other letter): m m m m+ + + m m m m+ m m m+ m+ + m + m m m To have the ommo startig ide of summatio m, write the first term i the first ad the third sums epliitly: m m m + + m+ m+ + m + + m m m m m m Now we a orgaize the equatio i oe power series m ( + ) + ( m+ )( m+ ) m+ + ( m+ ) m m We have that the power series is equal to zero. The aordig to the Idetity Theorem all oeffiiets should be equal to zero: ( + ) m+ m+ + m+ m,,... m+ m From whih we have (7) m+ m m+ m,,... (the reurree relatio) (8) Coeffiiets ad are ot defied by equatios (7-8) ad a be hose arbitrarily. The varyig the ide i equatios (7-8), we have: arbitrary arbitrary m 3 m m 3 3 m 4 m 5 5

Chapter V ODE V.4 Power Series Solutio Otober, 8 39 m m ( ) ( ) 3 + 4 + 4. Substitute the obtaied oeffiiets ito the power series y ( ) ( ) + 3 + 4 ( ) + ( ) + + 3 + 4 ( ) y ( ) 5.

7 Chapter V ODE V.4 Power Series Solutio Otober, 8 39 m m ( ) ( ) Substitute the obtaied oeffiiets ito the power series y ( ) ( ) ( ) + ( ) ( ) y ( ) 5. Geeral solutio i power series form: ( )! +!! + y ( ) y( ) ( )! ( ) + ( )!! +,, The last epressio is obtaied by maipulatio with the idies i the oeffiiets i order to obtai the simpler form of the result (this trasformatio is ot always obvious ad should ot eessarily be performed). The obtaied solutio i the form of two power series with the arbitrary oeffiiets ad is the geeral solutio of the ODE (6). There are o sigular poits of the equatio (6), therefore, both power series have a ifiite radius of overgee (it also a be ofirmed by the Ratio Test). Use Maple to seth the solutio urves. Note that i the Maple solutio (see Eample), the reurree equatio (8) is used diretly for evaluatio of the oeffiiets. Tris with fatorials: 3! 4 6 3! ( ) ( + ) ( 4 6 ) ( + ) ( + )!!

39 Chapter V ODE V.4 Power Series Solutio Otober, 8 V.4.3. THE METHOD OF FROBENIUS iterval of overgee ordiary poit overges sigular poit overges sigular poit If the liear ODE with variable oeffiiets

Sometimes a epasio about a sigular poit a have some advatage beause suh solutios a have a bigger radius of overgee.

Frobeius Theorem Theorem (power series solutio about the regular sigular poit ) Let the liear equatio y + p y + q y (3) haz a regular sigular poit, ad let p ( ) p + p + p +.

8 39 Chapter V ODE V.4 Power Series Solutio Otober, 8 V.4.3. THE METHOD OF FROBENIUS iterval of overgee ordiary poit overges sigular poit overges sigular poit If the liear ODE with variable oeffiiets has sigular poits the the radius of overgee of the power series solutio about the ordiary poit a be redued by the presee of sigular poits. Sometimes a epasio about a sigular poit a have some advatage beause suh solutios a have a bigger radius of overgee. The followig theorem provides the basis for the power series solutio about the regular sigular poit. For simpliity, we formulate this theorem for the regular sigular poit. Frobeius Theorem Theorem (power series solutio about the regular sigular poit ) Let the liear equatio y + p y + q y (3) haz a regular sigular poit, ad let p ( ) p + p + p +... p for R < (4a) q q + q + q +... q for < R (5a) Ferdiad Georg Frobeius ( ) Let r,r be the solutios of the idiial equatio r + ( p r ) + q () Re r Re r. ideed suh that The the liear ODE (3) has a geeral solutio + y ay by for R R mi R,R, ad liearly idepedet futios y ad y have the form: < <, where { }. If r r (do ot differ by a iteger) the + r y + r y d d. If r r (differ by a positive iteger) the + r y + r + y d y l d, a be zero 3. If r r (equal roots r r r) the + r y + r + y d y l

9 Chapter V ODE V.4 Power Series Solutio Otober, Eample Use the Frobeius method to fid a series solutio for the ODE y + y y () aroud the poit. Seth the solutio urves.. Poit is the oly sigular poit of the ODE (). Che if it is a regular sigular poit. Rewrite the equatio i the ormal form y + y y y + p y + q y p ( ) + is aalyti p q is aalyti q Therefore, is a regular sigular poit. Idiial equatio (): r + p r+ q r( r ) r + r+ roots r r r r That orrespods to Case of Frobeius Theorem:. The first solutio y a be foud i the form: + r + y + y + differetiate ad substitute ito Equatio () y m + m+ m + m m + m+ m + m hage of the idies m+ m+ m+ m+ m m m m m m m m m+ m+ + m+ m+

10 394 Chapter V ODE V.4 Power Series Solutio Otober, 8 write the zero terms epliitly m + m + m + + m + m + m+ m+ m+ m+ m m m m m m m m { } m + m m m m m+ m+ m+ + m+ { } m + m m m + m + m use the Idetity Theorem m m m + m + m m m+ + m m+ m m m reurree relatio m ay m m m is absorbed by ! y ! 3! 4! 5! ( ) ( + ) + st solutio! y 3. The seod solutio (ase of Frobeius Theorem) Substitutio of the trial form (ase ) ito equatio: multiply by ad add terms + y d y l y d y l y + + y ( ) d + y l + y + y y d + y l + y y + d + y l + y y l y d y l

11 Chapter V ODE V.4 Power Series Solutio Otober, d + d d d + y l + y l + y y + y y l y y l d + d d d + y y + y + y + y y y l beause y is a solutio d + d d d + y y+ y y y ( ) ( + ) + reall the st solutio! ( ) ( + ) + differetiate ad substitute ito the equatio! ( ) ( ) ( ) + ( + ) ( + ) d + d d d + +!!! m + m + m + m m m m m m m m+ m+ ( ) ( ) ( ) + ( + ) ( + ) + + +!!! d d d d ( ) ( ) ( ) m+ m m+ m m m m m m m ( + ) m m m m m m m m m m m md md m d d m! m! m! write terms for m ad m epliitly: m d d m + d+ d d d+ + ( ) ( ) ( ) m+ m m+ m m m m m m m ( + ) m m m m! m! m! m m m m m m m m md md m d d for m d d for m d + d d d + the the equatio beomes: m m m m ( d d) ( m ) mdm+ mdm ( m ) dm+ dm ombie i oe summatio m m m m

12 396 Chapter V ODE V.4 Power Series Solutio Otober, 8 { } m m+ m m m+ m m+ d + m d { } m m+ m m m+ m d + m d m+ m dm+ + m dm use the Idetity Theorem m+ m m+ d + d d dm + m+ m reurree formula for m d d d ay (but aot be hose as d ) d ay hoose d the d + m+ m dm m the the seod solutio beomes + d d + d d d d( ) y d y l or it a be simply writte as y ( ) The seod solutio of () 4. Alterative solutio for y ( ). Beause the seod solutio is i a simple losed form y ( ) we a use the redutio formula (Setio 5., Equatio (3)) to obtai the other liear idepedet solutio as: y a d a e y y d e ( ) ( ) d d e ( ) d d + l e d ( ) e d e ( ) (table itegral)

13 Chapter V ODE V.4 Power Series Solutio Otober, Let us see if we a retrieve this solutio from the power-series solutio. y ( ) We derived i the first part: ( ) ( + )! + +! 3! 4! ! 3! 4! add ad subtrat term e ( ) + + +! 3! 4! ( ) + e + e This solutio a be verified by diret substitutio ito ODE (). So, the geeral solutio of the ODE () a be writte as y y + y + e + ( )( ) e + + reame arbitrary ostats a ( ) + be The the Geeral Solutio of equatio () a be writte i losed form: y + e for all 5. Solutio urves for differet values of ad

14 398 V.4.4 Chapter V ODE V.4 Power Series Solutio Otober, 8 Taylor series solutio + + y( ) y p y q y y Loo for solutio i the form of the Taylor series about : iv y y 3 y 4 y( ) y( ) + y ( ) ( ) ! 3! 4! The first two oeffiiets are from the iitial oditios: y y The third oeffiiet a be foud from the give differetial equatio rewritte as, the evaluate y ( ) p( ) y ( ) q( ) y( ) y p y q y To fid the et oeffiiets, differetiate the equatio ad evaluate it at : y p y q y y... iv iv y y y... ad so o The with the foud ( ) y ostrut the Taylor series solutio iv y y 3 y 4 y( ) y( ) + y ( ) ( ) ! 3! 4!

15 Chapter V ODE V.4 Power Series Solutio Otober, Eample 3 Fid the first si terms of the power series solutio of the differetial equatio os y + y y subjet to iitial oditios: y, y 3. y

16 4 Chapter V ODE V.4 Power Series Solutio Otober, 8 V.4.5. POWER SERIES SOLUTION WITH MAPLE. Fid the solutio of the differetial equatio y + y + y (see Eample ) > restart; > M:4; M : 4 > []; > []; > []:-[]; : > for m from to M do [m+]:-[m]/(m+) od; 3 : 4 : 3 5 : 8 6 : 5 > m:'m': > y():sum([m]*^m,m..m+); y( ) : > y():ollet(y(),{[],[]}); y( ) : > f:subs({[]i,[]j},y()); f : i j > p:{seq(seq(f,i-4..4),j-4..4)}: > plot(p, ,olorbla);

17 Chapter V ODE V.4 Power Series Solutio Otober, 8 4. Solutio of IVP: y(), y'() > E:subs(,y()); > yp():diff(y(),); 5 E : yp( ) : > E:subs(,yp()); E : > solve({e,e},{[],[]}); {, } 8 7 > y():subs({[]-45/8,[]6/7},y()); y( ) : > plot(y(),-..);

18 4 Chapter V ODE V.4 Power Series Solutio Otober, 8 3. Power Series Solutio with the POWERSERIES paage: Fid the solutio of the differetial equatio 3 y y about the ordiary poit > restart; > with (powseries): > ODE:diff(y(),$)-^3*y(); d ODE : y( ) 3 y( ) d Fid the power series solutio with error of order O(^5): Covert to polyomial: > y():powsolve(ode); y( ) : pro( powparm )... ed pro > y():tpsform(y(),,5); C C C C y( ) : C + C O( 5 ) > y():overt(y(),polyom); y( ) : C + C C 5 3 C 6 8 C 33 C Plot solutio urves: > f:subs({ci,cj},y()); f : i + j i 5 3 j 6 8 i 33 j > p:{seq(seq(f,i-..),j-..)}: > plot(p,-3..,y-5..5,olorbla);

19 Chapter V ODE V.4 Power Series Solutio Otober, 8 43 V.4.6. REVIEW QUESTIONS. What is a aalytial futio? EXERCISES. What equatios are solved i the form of a power series? 3. What is a ordiary poit? 4. What is a sigular poit? 5. What is a regular sigular poit? 6. What is the radius of overgee of a power series solutio about a ordiary poit? 7. What are the mai steps i fidig a power series solutio about a ordiary poit? 8. What is the Method of Frobeius? 9. What is the idiial equatio?. What ases for the roots of the idiial equatio are osidered i the Frobeius Theorem?. What is the form of oe solutio whih a be foud for all three ases?. Why a it be advatageous to fid a solutio about a sigular poit? ) Give the power series epasios of ad their radius of overgee. ) Fid the iterval of overgee of the power series:! a) b) ( ) ) d) + 3 3) Fid the sigular poits of the equatios a) y y y ) + + b) λ d) y 3 y 4) Show that q( ) 3 ( ) 3 y + y y y y + y is aalyti at. i ad determie ( ) ( + 3 ) 5) Usig the power series method or the method of Frobeius, fid the geeral solutio of the followig differetial equatios: 3 a) y λ y f) y + y y b) y λ y g) y ( ) y y ) y y + y h) y + y y 3 d) y y i) y + y + y 4 e) y y y ;y y + + y y 6) Cosider the differetial equatio ) y y + y a) fid the geeral solutio of the give ODE i the form of a power series about the poit ; b) What is the radius of overgee of the obtaied power series solutio? ) Seth the solutio urves. 4 d) Fid the solutio subjet to the iitial oditios: y ; y 6.

20 44 Chapter V ODE V.4 Power Series Solutio Otober, 8

Explicit and closed formed solution of a differential equation. Closed form: since finite algebraic combination of. converges for x x0

Explicit and closed formed solution of a differential equation. Closed form: since finite algebraic combination of. converges for x x0 Chapter 4 Series Solutios Epliit ad losed formed solutio of a differetial equatio y' y ; y() 3 ( ) ( 5 e ) y Closed form: sie fiite algebrai ombiatio of elemetary futios Series solutio: givig y ( ) as