   Chapter 17.4, Problem 3E

Chapter
Section
Textbook Problem

Use power series to solve the differential equation.3. y' = x2y

To determine

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

Explanation

Given data:

The differential equation is,

yx2y=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)

Multiply x2 on both sides of equation (2).

x2y=n=0cnxn+2

x2y=n=2cn2xn (4)

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

n=0(n+1)cn+1xnn=2cn2xn=0

Rewrite the expression,

c1+2c2x+n=2[(n+1)cn+1cn2]xn=0 (5)

From equation (5), equating coefficients gives,

c1=0 and c2=0

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

(n+1)cn+1cn2=0 (6)

Re-arrange the equation.

cn+1=cn2(n+1)

cn+1=cn2(n+1),n=2,3, (7)

Equation (7) is the recursion relation.

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

Substitute 2 for n in equation (7),

cn+1=cn2(n+1)c2+1=c22(2+1)c3=c03

c3=c03 (8)

Substitute 3 for n in equation (7),

c3+1=c32(3+1)c4=c14

Substitute 0 for c1 ,

c4=04

c4=0 (9)

Substitute 4 for n in equation (7),

c4+1=c42(4+1)c5

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

In problems 37-44, perform the indicated operations and simplify. 40.

Mathematical Applications for the Management, Life, and Social Sciences

Find dy/dx by implicit differentiation. 9. x2x+y=y2+1

Single Variable Calculus: Early Transcendentals

Proof Prove Property 3 of Theorem 2.1. (You may use Property 3 of Theorem 2.2.)

Calculus: Early Transcendental Functions (MindTap Course List)

True or False: The x-intercepts are the values of x for which f(x) = 0.

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

The graph of x = cos t, y = sin2 t is:

Study Guide for Stewart's Multivariable Calculus, 8th

In Exercises 1320, plot the point on a set of coordinate axes. (52,32)

Finite Mathematics for the Managerial, Life, and Social Sciences

Define the concept of external validity and a threat to external validity.

Research Methods for the Behavioral Sciences (MindTap Course List) 