   Chapter 17.4, Problem 7E

Chapter
Section
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,

(x1)y+y=0 (1)

Consider the expression for y(x) .

y(x)=n=0cnxn (2)

Differentiate equation (2) with respect to t.

y(x)=n=1ncnxn1

y(x)=n=0(n+1)cn+1xn (3)

Differentiate equation (3) with respect to t.

y(x)=n=0(n+2)(n+1)cn+2xn (4)

Multiply (x1) with equation (4).

(x1)y(x)=(x1)n=0(n+2)(n+1)cn+2xn=n=0(n+2)(n+1)cn+2xn+1n=0(n+2)(n+1)cn+2xn

(x1)y(x)=n=1n(n+1)cn+1xnn=0(n+2)(n+1)cn+2xn (5)

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

(x1)y(x)=n=0n(n+1)cn+1xnn=0(n+2)(n+1)cn+2xn (6)

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

n=0n(n+1)cn+1xnn=0(n+2)(n+1)cn+2xn+n=0(n+1)cn+1xn=0n=0[n(n+1)cn+1(n+2)(n+1)cn+2+(n+1)cn+1]xn=0

n=0[(n+1)2cn+1(n+2)(n+1)cn+2]xn=0 (7)

Equation (7) is true when the coefficients are 0

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