   Chapter 17, Problem 17RE

Chapter
Section
Textbook Problem

Use power series to solve the initial-value problem y" + xy'  + y = 0 y(0) = 0 y'(0) = 1

To determine

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

Explanation

Given data:

The differential equation is,

y+xy+y=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)

Differentiate equation (3) with respect to t.

y(x)=n=0n(n1)cnxn2

y(x)=n=0(n+2)(n+1)cn+2xn (4)

Substitute equations (2), (3) and (4) in (1),

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

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

From equation (5), equating coefficients gives, and xn are 0. Therefore, the required expression,

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

Re-arrange the equation.

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

cn+2=cn(n+2),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).

Substitute 0 for n in equation (6),

c0+2=c0(0+2)

c2=c0(2) (7)

Write the given condition.

y(0)=0

Consider the expression.

y(0)=c0 (8)

Substitute 0 for c0 in equation (7)

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

Solve the equations in Exercises 126. x43x3=0

Finite Mathematics and Applied Calculus (MindTap Course List)

In Exercises 6372, evaluate the expression. 65. 12+41612

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Evaluate 11(x2x3)dx a) 32 b) 56 c) 12 d) 23

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 