Chapter 17.4, Problem 3E

### 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.3. y' = x2y

To determine

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

Explanation

Given data:

The differential equation is,

yâ€²âˆ’x2y=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)

Multiply âˆ’x2 on both sides of equation (2).

âˆ’x2y=âˆ’âˆ‘n=0âˆžcnxn+2

âˆ’x2y=âˆ’âˆ‘n=2âˆžcnâˆ’2xn (4)

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

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

Rewrite the expression,

c1+2c2x+âˆ‘n=2âˆž[(n+1)cn+1âˆ’cnâˆ’2]xn=0 (5)

From equation (5), equating coefficients gives,

c1=0 and c2=0

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

(n+1)cn+1âˆ’cnâˆ’2=0 (6)

Re-arrange the equation.

cn+1=cnâˆ’2(n+1)

cn+1=cnâˆ’2(n+1),â€‰â€‰n=2,3,â‹…â‹…â‹… (7)

Equation (7) is the recursion relation.

Solve the recursion relation by substituting n=2,3â‹…â‹…â‹… in equation (7).

Substitute 2 for n in equation (7),

cn+1=cnâˆ’2(n+1)c2+1=c2âˆ’2(2+1)c3=c03

c3=c03 (8)

Substitute 3 for n in equation (7),

c3+1=c3âˆ’2(3+1)c4=c14

Substitute 0 for c1 ,

c4=04

c4=0 (9)

Substitute 4 for n in equation (7),

c4+1=c4âˆ’2(4+1)c5

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