Chapter 17.4, Problem 8E

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Chapter
Section

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

# Use power series to solve the differential equation.8. y" = xy

To determine

To solve: The differential equation by the use of power series.

Explanation

Given data:

The differential equation is,

yâ€³=xy (1)

Consider the expression for y(x) ,

y(x)=âˆ‘n=0âˆžcnxn (2)

Differentiate equation (2) with respect to t,

yâ€²(x)=âˆ‘n=1âˆžncnxnâˆ’1 (3)

Differentiate equation (3) with respect to t,

yâ€³(x)=âˆ‘n=2âˆžn(nâˆ’1)cnxnâˆ’2

yâ€³(x)=âˆ‘n=0âˆž(n+2)(n+1)cn+2xn (4)

Write the expression for âˆ’xy(x) .

âˆ’xy(x)=âˆ’âˆ‘n=0âˆžcnxn+1=âˆ’âˆ‘n=1âˆžcnâˆ’1xn

Re-arrange equation (1).

yâ€³âˆ’xy=0

Substitute âˆ’âˆ‘n=1âˆžcnâˆ’1xn for âˆ’xy and equation (4) in (5).

âˆ‘n=0âˆž(n+2)(n+1)cn+2xnâˆ’âˆ‘n=1âˆžcnâˆ’1xn=02c2+âˆ‘n=1âˆž(n+2)(n+1)cn+2xnâˆ’âˆ‘n=1âˆžcnâˆ’1xn=0

2c2+âˆ‘n=1âˆž[(n+2)(n+1)cn+2âˆ’cnâˆ’1]xn=0 (5)

Equation (5) is true when the coefficients of xn are 0. Therefore, the required expressions are,

2c2=0c2=0

And

(n+2)(n+1)cn+2âˆ’cnâˆ’1=0

Re-arrange the equation,

cn+2=cnâˆ’1(n+2)(n+1),â€‰â€‰forâ€‰n=0,1,2â‹…â‹…â‹… (6)

Equation (6) is the recursion relation.

Since c2=0 , c3n+2=0 for n=0,1,2,â‹…â‹…â‹… .

Solve the recursion relation by substituting n=0,1,2,3â‹…â‹…â‹… in equation (6)

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