   Chapter 17.4, Problem 1E

Chapter
Section
Textbook Problem

Use power series to solve the differential equation.1. y' – y = 0

To determine

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

Explanation

Given data:

The differential equation is,

yy=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 (2), (3) and equation (1),

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

Substitute n+1 for n in the first sum of equation (4),

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

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

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

(n+1)cn+1cn=0

Re-arrange the equation.

(n+1)cn+1=+cn

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

Equation (6) is the recursion relation.

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

Substitute 0 for n in equation (6),

cn+1=cn(n+1)c1=c0(0+1)c1=c0

Substitute 1 for n in equation (6),

c1+1=c1(1+1)c2=c12

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