   Chapter 17.4, Problem 11E

Chapter
Section
Textbook Problem

Use power series to solve the differential equation.11. y" + x2y' + xy = 0, y(0) = 0, y'(0) = 1

To determine

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

Explanation

Given data:

The differential equation is,

y+x2y+xy=0 (1)

With y(0)=0 and y(0)=1 .

Consider the expression for y(x) .

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

Multiply x on both sides equation (2),

xy=xn=0cnxn

xy=n=0cnxn+1 (3)

Differentiate equation (2) with respect to t.

y(x)=n=1ncnxn1 (4)

Multiply x2 on both sides equation (4).

x2y=x2n=1ncnxn1

x2y=n=0ncnxn+1 (5)

Differentiate equation (4) with respect to t.

y(x)=n=2n(n1)cnxn2

y(x)=n=1(n+3)(n+2)cn+3xn+1

y(x)=2c2+n=0(n+3)(n+2)cn+3xn+1 (6)

Substitute equations (3), (5), and (6) in (1),

2c2+n=0(n+3)(n+2)cn+3xn+1+n=0ncnxn+1+n=0cnxn+1=0

2c2+n=0[(n+3)(n+2)cn+3+ncn+cn]xn+1=0 (7)

Equation (7) is true when the coefficients are 0. Therefore, the expressions are,

c2=0

And

(n+3)(n+2)cn+3+ncn+cn=0

Re-arrange the equation.

cn+3=ncncn(n+3)(n+2)

cn+3=(n+1)cn(n+3)(n+2),n=0,1,2 (8)

Equation (8) is the recursion relation

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