Solutions 3.2-Page 215

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1 Solutios.-Page Problem Fid the geeral solutios i powers of of the differetial equatios. State the reurree relatios ad the guarateed radius of overgee i eah ase. ) Substitutig,, ad ito the differetial equatio ields ) ) ) Simplifig further, ) ) The first ad third summatios a start at ad o additioal ozero terms will be added. However, the seod summatio must be rewritte to start at suh that the idetit priiple a be used. ) ) ) Usig the idetit priiple ad summig oeffiiets ields [ ] [ ] ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) So the reurree formula is From this reurree formula, it is evidet that oeffiiets with eve idies,, ) are equal to, ad oeffiiets with odd idies,, ) are equal to.

2 Therefore Epadig the summatios gives [ ] [ [ ] [ ] ] The first summatios math the form of if the variable is istead of. So The otatio of the tet is to write a differetial equatio as. For this problem,, whih has a sigular poit at, -. The distae from oe of these poits to a is, so the guarateed radius of overgee is at least. ) ) ) C B A ) A ρ

3 Problem Fid the geeral solutios i powers of of the differetial equatios. State the reurree relatios ad the guarateed radius of overgee i eah ase. Substitutig,, ad ito the differetial equatio ields ) ) Simplifig further, ) The seod summatio a start at ad o additioal ozero terms will be added. The first summatio must be rewritte to start at suh that the idetit priiple a be used. ) ) Usig the idetit priiple ad summig oeffiiets ields ) ) ) ) ) ) ) The reurree formula is ) The oeffiiets a be writte i terms of ad as follows:

4 : : : : : : 7 7 )! )!!! )! 7!!) 7!! For eve idies, For odd idies, Therefore, ) )! ) )!! )! )!! ) )! )!! The otatio of the tet is to write a differetial equatio as A ) B ) C ). For this problem, A ) whih has o sigular poit. ρ

5 Problem 7 Fid the geeral solutios i powers of of the differetial equatios. State the reurree relatios ad the guarateed radius of overgee i eah ase. ) 7 Substitutig,, ad ) ito the differetial equatio ields ) ) 7 Simplifig further, ) ) 7 The first ad third summatios a start at ad o additioal ozero terms will be added. However, the seod summatio must be rewritte to start at suh that the idetit priiple a be used. ) ) ) 7 Usig the idetit priiple ad summig oeffiiets ields ) ) 7 ) ) ) ) ) ) ) [ 8 ] ) ) The reurree formula is 7 7 [ 7 ) ] [ 7 ) ] ) ) ) ) ) ) )

6 The oeffiiets a be writte i terms of : : : : : 9: : 7 7: )) 7))) 7! )8) 9)8)7!)) 9! 9 9)9) )) ))9!) 9 ) ad as follows: 9!! )!! )! 9! Startig at, the patter that govers the odd ide oeffiiets is [ )!! ] ) 9 for )! Therefore [ )!! ] )! ) The otatio of the tet is to write a differetial equatio as A ) B ) C ). For this problem, A ), whih has a sigular poit at ±. The distae from oe of these poits to a is, so the guarateed radius of overgee is at least. ρ

7 Problem Fid a three-term reurree relatio for solutios of the form the first three ozero terms i eah of two liearl idepedet solutios.. The fid ) Substitutig ad ) ito the differetial equatio ields ) ) Simplifig further, ) The first summatio a be started at to get a start at i order to get a term. term. The third summatio must ) ) The ommo rage is, so the terms orrespodig to must be brought out. [ ) ) ] The idetit priiple implies that ad [ ) ) ]. The reurree formula for is ) ) The first liearl idepedet solutio a be obtaied b settig ad. Therefore /. The et oeffiiet is foud usig the reurree formula. : /

8 The et liearl idepedet solutio a be obtaied b settig ad. Therefore. The et oeffiiets are foud usig the reurree formula. : : / /

9 Problem 7 Solve the iitial value problem ) ; ), ) Determie suffiietl ma terms to ompute / ) aurate to four deimal plaes. Substitutig,, ad ) ito the differetial equatio ields ) ) Simplifig further, ) All of the summatios must be writte i terms of priiple. i order to use the idetit ) ) The ommo rage is, so the terms orrespodig to, must be brought out. [ ) ) ] The idetit priiple implies the followig: ) for ) ) ) ) From the give iitial oditios, ad. The remaiig oeffiiets are determied as follows:

10 / / : : : : : So 7 8 ) ) ) ) 7) / / / / / 9 / 7 / / / / 7 / / ). 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