Chapter 17, Problem 18RE

### 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 differential equation y" – xy' – 2y = 0

To determine

To solve: The differential equation by using power series.

Explanation

Given data:

The differential equation is,

yâ€³âˆ’xyâ€²âˆ’2y=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=0âˆžncnxnâˆ’1 (3)

Multiply x on both sides equation (3).

xyâ€²(x)=xâˆ‘n=0âˆžncnxnâˆ’1

xyâ€²(x)=âˆ‘n=0âˆžncnxn (4)

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 (5)

Substitute equations (4) in (1),

âˆ‘n=0âˆž(n+2)(n+1)cn+2xnâˆ’âˆ‘n=0âˆžncnxnâˆ’2(âˆ‘n=0âˆžcnxn)=0

âˆ‘n=0âˆž[(n+2)(n+1)cn+2âˆ’(n+2)cn]xn=0 (6)

Equation (6) is true when the coefficients are 0. Therefore, the expressions are,

(n+2)(n+1)cn+2âˆ’(n+2)cn=0

Re-arrange the equation.

cn+2=(n+2)cn(n+2)(n+1)

cn+2=cnn+1,â€‰â€‰n=0,1,2â‹…â‹…â‹… (7)

Equation (7) is the recursion relation.

Solve the recursion relation by substituting different n values in equation (7).

By recursion relation, consider the expression.

Substitute 0 for n in equation (7),

c0+2=c00+1c2=c0

Substitute 2 for n in equation (7),

c2+2=c22+1c4=c23

Substitute c0 for c2 ,

c4=c01Ã—3 (8)

Substitute 4 for n in equation (7),

c4+2=c44+1c6=c45

Substitute c21Ã—3 for c4 ,

c6=c21Ã—35=c21Ã—3Ã—5

Substitute c0 for c2 ,

c6=

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