Chapter 17.2, Problem 3E

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Chapter
Section

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Solve the differential equation or initial-value problem using the method of undetermined coefficients.3. 9y" + y = e2x

To determine

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

Explanation

Given data:

The differential equation is,

9yâ€³+y=e2x (1)

Formula used:

Write the expression for a second differential equation.

ayâ€³+byâ€²+cy=0 (2)

Consider the expression for an auxiliary equation.

ar2+br+c=0 (3)

Write the expression for the complex roots.

r=Î±Â±iÎ² (4)

Here,

Î± is the real part of the root, and

Î² is the imaginary part of the root.

Write the expression for the complementary solution of ayâ€³+byâ€²+cy=0 with complex roots.

yc(x)=eÎ±x(c1cosÎ²x+c2sinÎ²x) (5)

Write the expression for the particular solution yp(x) .

yp(x)=Ae2x (6)

Compare equation (1) and (2).

a=9b=0c=1

Substitute 9 for a, 0 for b, and 1 for c in equation (3),

(9)r2+(0)r+(1)=09r2+1=0r2=âˆ’19

r=Â±13i (7)

Compare equations (4) and (7).

Î±=0Î²=13

Substitute 0 for Î± and 13 for Î² in equation (5),

yc(x)=e0x(c1cos(x3)+c2sin(x3))

yc(x)=c1cos(x3)+c2sin(x3) (8)

Differentiate equation (6) with respect to x,

yâ€²p(x)=ddx

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