   Chapter 17.4, Problem 2E

Chapter
Section
Textbook Problem

1-11 Use power series to solve the differential equation.2. y' = xy

To determine

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

Explanation

Given data:

The differential equation is,

yxy=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)

Substitute equations (3) and (2) in (1),

n=1ncnxn1xn=0cnxn=0

n=1ncnxn1n=0cnxn+1=0 (4)

Substitute n+1 for n in the first sum and n1 for n in second sum of equation (4).

y(x)=n=0(n+1)c(n+1)xnn=1cn1xn=0c1+n=1(n+1)c(n+1)xnn=1cn1xn=0

c1+n=1[(n+1)c(n+1)cn1]xn=0 (5)

Equate coefficients from equation (5),

c1=0

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

(n+1)cn+1cn1=0 (6)

Re-arrange the equation.

cn+1=cn1(n+1)

cn+1=cn1(n+1),n=1,2 (7)

Equation (7) is the recursion relation.

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

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