BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Solutions

Chapter
Section
BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Use power series to solve the differential equation.

8. y" = xy

To determine

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

Explanation

Given data:

The differential equation is,

y=xy (1)

Consider the expression for y(x) ,

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

Differentiate equation (2) with respect to t,

y(x)=n=1ncnxn1 (3)

Differentiate equation (3) with respect to t,

y(x)=n=2n(n1)cnxn2

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

Write the expression for xy(x) .

xy(x)=n=0cnxn+1=n=1cn1xn

Re-arrange equation (1).

yxy=0

Substitute n=1cn1xn for xy and equation (4) in (5).

n=0(n+2)(n+1)cn+2xnn=1cn1xn=02c2+n=1(n+2)(n+1)cn+2xnn=1cn1xn=0

2c2+n=1[(n+2)(n+1)cn+2cn1]xn=0 (5)

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

2c2=0c2=0

And

(n+2)(n+1)cn+2cn1=0

Re-arrange the equation,

cn+2=cn1(n+2)(n+1),forn=0,1,2 (6)

Equation (6) is the recursion relation.

Since c2=0 , c3n+2=0 for n=0,1,2, .

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

Chapter 17 Solutions

Show all chapter solutions add
Sect-17.1 P-11ESect-17.1 P-12ESect-17.1 P-13ESect-17.1 P-14ESect-17.1 P-15ESect-17.1 P-16ESect-17.1 P-17ESect-17.1 P-18ESect-17.1 P-19ESect-17.1 P-20ESect-17.1 P-21ESect-17.1 P-22ESect-17.1 P-23ESect-17.1 P-24ESect-17.1 P-25ESect-17.1 P-26ESect-17.1 P-27ESect-17.1 P-28ESect-17.1 P-29ESect-17.1 P-30ESect-17.1 P-31ESect-17.1 P-32ESect-17.1 P-33ESect-17.1 P-34ESect-17.2 P-1ESect-17.2 P-2ESect-17.2 P-3ESect-17.2 P-4ESect-17.2 P-5ESect-17.2 P-6ESect-17.2 P-7ESect-17.2 P-8ESect-17.2 P-9ESect-17.2 P-10ESect-17.2 P-11ESect-17.2 P-12ESect-17.2 P-13ESect-17.2 P-14ESect-17.2 P-15ESect-17.2 P-16ESect-17.2 P-17ESect-17.2 P-18ESect-17.2 P-19ESect-17.2 P-20ESect-17.2 P-21ESect-17.2 P-22ESect-17.2 P-23ESect-17.2 P-24ESect-17.2 P-25ESect-17.2 P-26ESect-17.2 P-27ESect-17.2 P-28ESect-17.3 P-1ESect-17.3 P-2ESect-17.3 P-3ESect-17.3 P-4ESect-17.3 P-5ESect-17.3 P-6ESect-17.3 P-7ESect-17.3 P-8ESect-17.3 P-9ESect-17.3 P-10ESect-17.3 P-11ESect-17.3 P-12ESect-17.3 P-13ESect-17.3 P-14ESect-17.3 P-15ESect-17.3 P-16ESect-17.3 P-17ESect-17.3 P-18ESect-17.4 P-1ESect-17.4 P-2ESect-17.4 P-3ESect-17.4 P-4ESect-17.4 P-5ESect-17.4 P-6ESect-17.4 P-7ESect-17.4 P-8ESect-17.4 P-9ESect-17.4 P-10ESect-17.4 P-11ESect-17.4 P-12ECh-17 P-1RCCCh-17 P-2RCCCh-17 P-3RCCCh-17 P-4RCCCh-17 P-5RCCCh-17 P-1RQCh-17 P-2RQCh-17 P-3RQCh-17 P-4RQCh-17 P-1RECh-17 P-2RECh-17 P-3RECh-17 P-4RECh-17 P-5RECh-17 P-6RECh-17 P-7RECh-17 P-8RECh-17 P-9RECh-17 P-10RECh-17 P-11RECh-17 P-12RECh-17 P-13RECh-17 P-14RECh-17 P-15RECh-17 P-16RECh-17 P-17RECh-17 P-18RECh-17 P-19RECh-17 P-20RECh-17 P-21RE

Additional Math Solutions

Find more solutions based on key concepts

Show solutions add

Powers Evaluate the power, and write the result in the form a + bi. 47. i3

Precalculus: Mathematics for Calculus (Standalone Book)

In Exercises 516, evaluate the given quantity. log1616

Finite Mathematics and Applied Calculus (MindTap Course List)

In Exercises 7-22, evaluate the expression. 22. 62.561.961.4

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

Find the limit. 35. lims164ss16

Single Variable Calculus

Using the table from Exercise 5, sketch the graph of 2x+3y=12.

Elementary Geometry For College Students, 7e

The third partial derivative wxyz of the function w=4xyz+x3y2z+x3 .

Mathematical Applications for the Management, Life, and Social Sciences

For 0

Study Guide for Stewart's Multivariable Calculus, 8th

For what values of p does the series converge?

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