Chapter 17.4, Problem 5E

### 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.5. y" + xy' + y = 0

To determine

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

Explanation

Given data:

The differential equation is,

yâ€³+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 (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+1)(n+2)cn+2xn (4)

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

âˆ‘n=0âˆž(n+1)(n+2)cn+2xn+xâˆ‘n=1âˆžncnxnâˆ’1+âˆ‘n=0âˆžcnxn=0 (5)

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

(n+1)(n+2)cn+2+(n+1)cn=0

Re-arrange the equation.

cn+2=âˆ’(n+1)cn(n+2)(n+1),â€‰â€‰n=0,1,2â‹…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)(0+1)c2=âˆ’c02

Substitute 2 for n in equation (6),

c2+2=âˆ’3c2(2+2)(2+1)c4=âˆ’3c212

c4=âˆ’c24

Substitute âˆ’c02 for c2 .

c4=âˆ’âˆ’c024

c4=c02Ã—4 (7)

Substitute 4 for n in equation (6),

c4+2=âˆ’5c4(4+2)(4+1)c6=âˆ’5c430c6=âˆ’c46

Substitute c02Ã—4 for c4 ,

c6=c02Ã—46

c6=c02Ã—4Ã—6 (8)

Similarly, like equations (7) and (8), write the expression for c2n ,

c2n=(âˆ’1)nc02Ã—4â‹…â‹…â‹…â‹…â‹…2n

c2n=(âˆ’1)nc02nn! (9)

Substitute 1 for n in equation (6),

c1+2=âˆ’2c1(1+2)(1+1)c3=âˆ’2c16c3=âˆ’c13

Substitute 3 for n in equation (6),

c3+2=âˆ’4c3(3+2)(3+1)c5=âˆ’4c320c5=âˆ’c35

Substitute âˆ’c13 for c3

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