   Chapter 17.4, Problem 8E

Chapter
Section
Textbook Problem

Use power series to solve the differential equation.8. y" = xy

To determine

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

Explanation

Given data:

The differential equation is,

y=xy (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+2)(n+1)cn+2xn (4)

Write the expression for xy(x) .

xy(x)=n=0cnxn+1=n=1cn1xn

Re-arrange equation (1).

yxy=0

Substitute n=1cn1xn for xy and equation (4) in (5).

n=0(n+2)(n+1)cn+2xnn=1cn1xn=02c2+n=1(n+2)(n+1)cn+2xnn=1cn1xn=0

2c2+n=1[(n+2)(n+1)cn+2cn1]xn=0 (5)

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

2c2=0c2=0

And

(n+2)(n+1)cn+2cn1=0

Re-arrange the equation,

cn+2=cn1(n+2)(n+1),forn=0,1,2 (6)

Equation (6) is the recursion relation.

Since c2=0 , c3n+2=0 for n=0,1,2, .

Solve the recursion relation by substituting n=0,1,2,3 in equation (6)

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