Chapter 17.2, Problem 5E

### 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.5. y" – 4y' + 5y = e-x

To determine

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

Explanation

Given data:

The differential equation is,

yâ€³âˆ’4yâ€²+5y=eâˆ’x (1)

Formula used:

Write the expression for a second order differential equation.

ayâ€³+byâ€²+cy=0 (2)

Write 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)=Aeâˆ’x (6)

Write the expression to find the roots of quadratic equation.

r=âˆ’bÂ±b2âˆ’4ac2a (7)

Compare equation (1) and (2).

a=1b=âˆ’4c=5

Substitute 1 for a, â€“4 for b, and 5 for c in equation (3),

(1)r2+(âˆ’4)r+(5)=0r2âˆ’4r+5=0

Find the roots of auxiliary equation using equation (7).

Substitute 1 for a, â€“4 for b, and 5 for c in equation (7),

r=âˆ’(âˆ’4)Â±(âˆ’4)2âˆ’4(1)(5)2(1)=4Â±16âˆ’202=4Â±âˆ’42=4Â±i22

r=2Â±i (8)

Compare equations (4) and (8)

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