Chapter 17, Problem 17RE

### 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 initial-value problem y" + xy'  + y = 0 y(0) = 0 y'(0) = 1

To determine

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

Explanation

Given data:

The differential equation is,

yâ€²â€²+xyâ€²+y=0 (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

yâ€²(x)=âˆ‘n=0âˆž(n+1)cn+1xn (3)

Differentiate equation (3) with respect to t.

yâ€³(x)=âˆ‘n=0âˆžn(nâˆ’1)cnxnâˆ’2

yâ€³(x)=âˆ‘n=0âˆž(n+2)(n+1)cn+2xn (4)

Substitute equations (2), (3) and (4) in (1),

âˆ‘n=0âˆž(n+2)(n+1)cn+2xn+xâˆ‘n=0âˆž(n+1)cn+1xn+âˆ‘n=0âˆžcnxn=0

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

From equation (5), equating coefficients gives, and xn are 0. Therefore, the required expression,

(n+2)cn+2+cn=0

Re-arrange the equation.

(n+2)cn+2+cn=0cn+2=âˆ’cn(n+2)

cn+2=âˆ’cn(n+2),â€‰â€‰n=0,1,2,3,â‹…â‹…â‹… (6)

Equation (6) is the recursion relation.

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

Substitute 0 for n in equation (6),

c0+2=âˆ’c0(0+2)

c2=âˆ’c0(2) (7)

Write the given condition.

y(0)=0

Consider the expression.

y(0)=c0 (8)

Substitute 0 for c0 in equation (7)

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