   Chapter 17.4, Problem 10E

Chapter
Section
Textbook Problem

Use power series to solve the differential equation.10. y" + x2y = 0, y(0) = 1, y'(0) = 0

To determine

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

Explanation

Given data:

The differential equation is,

y+x2y=0 (1)

with y(0)=1,y(0)=1 .

Consider the expression for y(x) ,

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

Calculate x2y by using equation (2),

x2y=x2n=0cnxn=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=1n(n1)cnxn2=n=2(n+4)(n+3)cn+4xn+2

y(x)=2c2+6c3x+n=0(n+4)(n+3)cn+4xn+2 (4)

Substitute n=0cnxn+2 for x2y and equation (4) in equation (1),

2c2+6c3x+n=0(n+4)(n+3)cn+4xn+2+n=0cnxn+2=0

2c2+6c3x+n=0[(n+4)(n+3)cn+4+cn]xn+2=0 (5)

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

2c2=0c2=0

6c3=0c3=0

And

(n+4)(n+3)cn+4+cn=0

Re-arrange the equation.

cn+4=cn(n+4)(n+3),n=0,1,2 (6)

Equation (6) is the recursion relation.

Solve the recursion relation by substituting different n values in equation (6).

By recursion relation, consider the expression

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