   Chapter 17, Problem 5RE

Chapter
Section
Textbook Problem

Solve the differential equation.5.  d 2 y d x 2 − 4 d y d x + 5 y = e 2 x

To determine

To solve: The differential equation of d2ydx24dydx+5y=e2x .

Explanation

Given data:

The differential equation is,

d2ydx24dydx+5y=e2x

y4y+5y=e2x (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)=Ae2x (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)

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

A sample of n = 9 scores has X = 108. What is the sample mean?

Essentials of Statistics for The Behavioral Sciences (MindTap Course List)

In Exercises 31-34, evaluate h(2), where h = g f. 31. f(x) = x2 + x + 1; g(x) = x2

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

True or False: Partial fractions may be used for .

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