Chapter 17.4, Problem 7E

### 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.7. (x – 1)y" + y' = 0

To determine

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

Explanation

Given data:

The differential equation is,

(xâˆ’1)yâ€³+yâ€²=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=0âˆž(n+2)(n+1)cn+2xn (4)

Multiply (xâˆ’1) with equation (4).

(xâˆ’1)yâ€³(x)=(xâˆ’1)âˆ‘n=0âˆž(n+2)(n+1)cn+2xn=âˆ‘n=0âˆž(n+2)(n+1)cn+2xn+1âˆ’âˆ‘n=0âˆž(n+2)(n+1)cn+2xn

(xâˆ’1)yâ€³(x)=âˆ‘n=1âˆžn(n+1)cn+1xnâˆ’âˆ‘n=0âˆž(n+2)(n+1)cn+2xn (5)

Since âˆ‘n=1âˆžn(n+1)cn+1xn=âˆ‘n=0âˆžn(n+1)cn+1xn , the equation (5) is written as,

(xâˆ’1)yâ€³(x)=âˆ‘n=0âˆžn(n+1)cn+1xnâˆ’âˆ‘n=0âˆž(n+2)(n+1)cn+2xn (6)

Substitute equations (3), and (6) in (1),

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

âˆ‘n=0âˆž[(n+1)2cn+1âˆ’(n+2)(n+1)cn+2]xn=0 (7)

Equation (7) is true when the coefficients are 0

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