   Chapter 17.2, Problem 5E

Chapter
Section
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,

y4y+5y=ex (1)

Formula used:

Write the expression for a second order differential equation.

ay+by+cy=0 (2)

Write the expression for an auxiliary equation.

ar2+br+c=0 (3)

Write the expression for the complex roots.

r=α±iβ (4)

Here,

α is the real part of the root, and

β is the imaginary part of the root.

Write the expression for the complementary solution of ay+by+cy=0 with complex roots.

yc(x)=eαx(c1cosβx+c2sinβx) (5)

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

yp(x)=Aex (6)

Write the expression to find the roots of quadratic equation.

r=b±b24ac2a (7)

Compare equation (1) and (2).

a=1b=4c=5

Substitute 1 for a, –4 for b, and 5 for c in equation (3),

(1)r2+(4)r+(5)=0r24r+5=0

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

Substitute 1 for a, –4 for b, and 5 for c in equation (7),

r=(4)±(4)24(1)(5)2(1)=4±16202=4±42=4±i22

r=2±i (8)

Compare equations (4) and (8)

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