   Chapter 17.2, Problem 8E

Chapter
Section
Textbook Problem

Solve the differential equation or initial-value problem using the method of undetermined coefficients.8. y" – y' = xe2x, y(0) = 0, y'(0) = 1

To determine

To solve: The differential equation by the method of undetermined coefficients.

Explanation

Given data:

The differential equation is,

yy=xe2x (1)

with y(0)=0,andy(0)=1 .

Consider the auxiliary equation is,

r21=0

r2=1 (2)

Roots of equation (2) are,

r=±1

Write the expression for the complementary solution,

yc(x)=c1ex+c2ex

The particular solution yp(x) is,

yp(x)=(Ax+B)e2x (3)

Differentiate equation (3) with respect to x,

yp(x)=ddx((Ax+B)e2x)

yp(x)=(2Ax+A+2B)e2x (4)

Differentiate equation (3) with respect to x,

yp(x)=ddx((2Ax+A+2B)e2x)

yp(x)=(4Ax+4A+4B)e2x (5)

Substitute equations (3) and (5) in equation (1),

(4Ax+4A+4B)e2x(Ax+B)e2x=xe2x

(3Ax+4A+3B)e2x=xe2x

3Ax+4A+3B=x (6)

Differentiate the equation (6) with respect to x.

ddx(3Ax+4A+3B)=ddx(x)ddx(3Ax)+ddx(4A)+ddx(3B)=ddx(x)3A+0+0=13A=1

Simplify the equation.

A=13

Substitute 0 for x in equation (6),

3A(0)+4A+3B=0

4A+3B=0 (7)

Substitute 90° for x in equation (6),

(4A2B)cos90°+(2A+4B)sin90°=sin90°(4A2B)(0)+(2A+4B)(1)=(1)

2A+4B=1 (8)

Substitute 13 for A in equation (8)

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 