   Chapter 17.4, Problem 9E

Chapter
Section
Textbook Problem

Use power series to solve the differential equation.9. y" – xy' – y = 0, y(0) = 1, y'(0) = 0

To determine

To solve: The differential equation by using power series.

Explanation

Given data:

The differential equation is,

yxyy=0,y(0)=1,y(0)=0 (1)

Consider the expression for y(x) .

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

Differentiate equation (2) with respect to t.

y(x)=n=1ncnxn1 (3)

Differentiate equation (3) with respect to t.

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

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

Consider the expression for xy(x) .

xy(x)=xn=1ncnxn1=n=1ncnxn

xy(x)=n=0ncnxn (5)

Substitute equations (2), (4) and (5) in (1),

n=0(n+1)(n+2)cn+2xnn=0ncnxn+n=0cnxn=0

y(x)=n=0[(n+1)(n+2)cn+2ncncn]xn=0 (6)

Equation (6) is true, when the coefficients of xn are 0. Therefore, the required expression is,

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

Re-arrange the equation.

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

cn+2=cn(n+2),n=0,1,2 (7)

Equation (7) is the recursion relation.

Consider the following condition.

y(0)=1

Substitute 0 for x in equation (6).

y(0)=n=0cn0n

y(0)=c0+0+0 (8)

From the expression, the value of c0 is,

c0=1

Solve the recursion relation by substituting n=0,1,2,3 in equation (7)

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