   Chapter 17.4, Problem 4E

Chapter
Section
Textbook Problem

Use power series to solve the differential equation.4. (x – 3)y' + 2y = 0

To determine

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

Explanation

Given data:

The differential equation is,

(x3)y+2y=0 (1)

Consider the expression for y(x) .

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

Differentiate equation (2) with respect to t.

y(x)=n=1ncnxn1

y(x)=n=0(n+1)cn+1xn (3)

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

(x3)n=0(n+1)cn+1xn+2n=0cnxn=0

Re-write the expression.

n=0(n+1)cn+1xn+13n=0(n+1)cn+1xn+2n=0cnxn=0

Re-write the expression for xn .

n=1ncnxnn=03(n+1)cn+1xn+n=02cnxn=0{n=1ncnxn=n=0ncnxn}

n=0[(n+2)cn3(n+1)cn+1]xn=0 (4)

From equation (4), equating coefficients gives, and xn are 0. Therefore, the required expression is,

(n+2)cn3(n+1)cn+1=0 (5)

Re-arrange the equation.

3(n+1)cn+1=(n+2)cncn+1=(n+2)cn3(n+1)

cn+1=(n+2)cn3(n+1),n=0,1,2,3, (6)

Equation (6) is the recursion relation.

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

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Find more solutions based on key concepts 