Chapter 17.4, Problem 1E

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Chapter
Section

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
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,

yâ€²âˆ’y=0 (1)

Consider the expression for y(x) .

y(x)=âˆ‘n=0âˆžcnxn (2)

Differentiate equation (2) with respect to t,

yâ€²(x)=âˆ‘n=1âˆžncnxnâˆ’1 (3)

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

âˆ‘n=0âˆžncnxnâˆ’1âˆ’âˆ‘n=0âˆžcnxn=0 (4)

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

âˆ‘n=0âˆž(n+1)cn+1xnâˆ’âˆ‘n=0âˆžcnxn=0

âˆ‘n=0âˆž[(n+1)cn+1âˆ’cn]xn=0 (5)

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

(n+1)cn+1âˆ’cn=0

Re-arrange the equation.

(n+1)cn+1=+cn

cn+1=cn(n+1),â€‰â€‰n=0,1,2â‹…â‹…â‹… (6)

Equation (6) is the recursion relation.

Solve the recursion relation by substituting n=0,1,2,3â‹…â‹…â‹… 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

### 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

#### Let f(x) = 3x2 + 5x 2. Find: a. f(2) b. f(a + 2) c. f(2a) d. f(a + h)

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

Elementary Technical Mathematics

#### In Problems 13-26, graph the solution of each system of inequalities. 26.

Mathematical Applications for the Management, Life, and Social Sciences

#### True or False: is a convergent series.

Study Guide for Stewart's Multivariable Calculus, 8th

#### Finding a Limit In Exercises 47-62, find the limit. limx0x+55x

Calculus: Early Transcendental Functions (MindTap Course List)

#### limx5x225x29x+20= a) 1 b) 10 c) d) does not exist

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