Explicit and closed formed solution of a differential equation. Closed form: since finite algebraic combination of. converges for x x0
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1 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 a ifiite series ivolvig ostat time powers of (Def 4.4) Power Series a overges for E6:!, whih overges for but diverges for ay
2 (Thm 4.5) overges for Suppose a series overges absolutely for all suh that.the the : Three possibilities for overgee of a. overge oly for. overge for all real umber 3. positive umber r, alled the radius of overgee of the series, suh that a diverges if r iterval of overgee. (Thm 4.6) overges if. r, r Suppose b for,,,... ad that r, is alled the ope lim b b L The b overges absolutely if L ad diverges if L. If L, the this test allows o olusio
3 Taylor ad Malauri Epasios i some iterval r, r f ( ) a f a f ' a f '' a (3) f 3 a3 ( k ) f k k a k k! ( ) ( k ) ak f This umber is alled the k th Taylor oeffiiet of f at! f The series is the Taylor series for f ( ) about If, the Taylor epasio is about ad is ofte alled a Malauri epasio e si!! are Malauri epasios 3
4 4. Power Series Solutios of Iitial Value Problems (Def 4.) Aalyti Futio: A futio f is aalyti at if f ( ) has a power series represetatio i some ope iterval about : f ( ) a ( ) i some iterval ( r, r). r is the radius of overgee ( 收斂半徑 ) (Thm 4.) Let p ad q be aalyti at. The the iitial value problem y ' py ( ) q ( ); y ( ) y has a solutio that is aalyti at That meas oeffiiets are aalyti at has a aalyti solutio at y ( ) a( ) ( ).! ( ) a y 4
5 E63: y ' e y ; y() 4 [ 解 ]: 5
6 (Thm 4.) Let p, q, ad f be aalyti at. The the iitial value problem y" py ( ) ' qy ( ) f( ) ; y ( ) A, y'( ) B has a uique solutio that is also aalyti at E64: y" y' e y 4; y(), y '() 4 [ 解 ]: 6
7 Eerise V: I eah problem, fid the first five ozero terms of the power series solutio of the iitial value problem, about the poit where the iitial oditios are give.. y" y' y ; y(), y '() { y } 3 6. y" y ; y() 3, y '() } { y 3 3. y" e y' y ; y() 3, y '() { y 34 } 6 3 7
8 4. Power Series Solutios Usig Reurree Relatios E65: y '' y,we wat a solutio epaded about [ 解 ]: y a y' a y'' a y ' begis at, y '' begis at Substitute yy, '' ito the differetial equatio '' y y a a a m m a m ( m, m ) m a am m ( m, m ) (m ) a a m a 3a a a 3 m 8
9 a a3 a a for =, 3,. a a a a a 43 4 a a a 54 5 for =, 3,. y a a a a a a a a a y, a y ' 9
10 E66: y '' y' 4 y [ 解 ]:
11 Eerise W: I eah problem, fid the reurree relatio ad use it to geerate the first five terms of the Malauri series of the geeral solutio.. y' y { y } a. y'' y' y { y ay ay, a y, a y' y , 5 3 y } 3. y y y '' ' ( ) { y a a a y, a y' }
12 4.3 Sigular Poits ad the Method of Forbeius '' ' P y Q y R y F ' y'' p y q y f (Def 4.) Ordiary ad Sigular Poits: is a ordiary poit of equatio '' ' if P y Q y R y F P ad Q/ P, R/ P,ad / is a sigular poit of equatio '' ' F P are aalyti at P y Q y R y F if is ot a ordiary poit E67: [ 解 ]: 3 y y y '' 5 ' 3
13 Oly fous o homogeeous equatio P y'' Q y' R y (Def 4.3) Regular ad Irregular Sigular Poits: is a regular sigular poit of equatio Q ad P P y'' Q y' R y if is a sigular poit, ad the futios R are aalyti at P. A sigular poit that is ot regular is said to be a irregular sigular poit E68: [ 解 ]: 3 y y y '' 5 ' 3 3
14 For regular sigular poit at y r is alled Forbeius series A Forbeius series eed ot be a power series, r may be egative or a oiteger Begis with ( ) r, whih is ostat oly if r y' r r This summatio for y ' begis at zero. Also true for y '' y'' r r r E69: y'' y' y [ 解 ]: 4
15 5
16 4.4 Seod Solutios (Thm 4.4) A seod solutio i the method of Frobeius: If ( r r ) is ot a iteger, two liearly idepedet Frobeius solutios: y ( ) r * r ad y ( ), * y ad y form a fudametal set o some iterval (, r) or (-r, ) 6
17 Eerise X: Fid all of the sigular poits ad lassify eah sigular poit as regular or irregular.. 3 y'' 4 6 y' y { Sigular poits are ad 3. is regular, 3 is regular} I eah problem, (a) show that zero is a regular sigular poit of the differetial equatio, (b) fid ad solve the idiial equatio, () determie the reurree relatio ad (d) use the results of (b) ad () to fid the first five ozero terms of two liearly idepedet Frobeius solutios.. 4 y'' y' y {(a) Q P ad R 4 are both aalyti at P so, is a regular sigular poit. (b) r r with roots r ad r () The reurree relatio is r r, 7
18 (d) Usig y r gives Usig r gives solutio 3. Q {(a) P 9 y'' y (b) y ad 9r 9r! }! R P ad oe solutio ad a seod, both aalyti at 9 with roots r 3 ad r 3 () 4, 3 3 r r (d) Usig r 3 gives ad 3 4 y 3! 47 3 Whereas r 3 gives ad 3 4 y 3! 58 3 } 8
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