   Chapter 17.1, Problem 20E

Chapter
Section
Textbook Problem

Solve the initial-value problem.20. 3y" – 2y' – y = 0, y(0) = 0, y'(0) = –4

To determine

To solve: The initial-value problem for differential equation 3y2yy=0 , y(0)=0 , y(0)=4 .

Explanation

Formula used:

Write the expression for differential equation.

ay+by+cy=0 (1)

Write the expression for auxiliary equation.

ar2+br+c=0 (2)

Write the expression for general solution of ay+by+cy=0 .

y=c1er1x+c2er2x (3)

Here,

r1 and r2 are the roots of auxiliary equation.

Write the required differential formulae to evaluate the differential equation.

ddxenx=nenx

Consider the differential equation as follows.

3y2yy=0 (4)

Compare equation (1) and (4).

a=3b=2c=1

Find the auxiliary equation.

Substitute 3 for a , 2 for b and 1 for c in equation (2),

(3)r2+(2)r+(1)=03r22r1=0(3r+1)(r1)=0

Simplify the equation as follows.

3r+1=0r=13r1=0r=1

Consider the value of r1 and r2 as follows.

r1=13r2=1

Find the general solution of 3y2yy=0 using equation (3).

Substitute 13 for r1 and 1 for r2 in equation (3),

y=c1e13x+c2ex

y=c1ex3+c2ex (5)

Modify equation (5) as follows.

y(x)=c1ex3+c2ex (6)

Find the value of y(0)

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

In Problems 81 and 82, find the fifth derivatives. 82.

Mathematical Applications for the Management, Life, and Social Sciences

∫x−4dx = −4x−5 + C

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

Given that for all x, a power series for is:

Study Guide for Stewart's Multivariable Calculus, 8th

In Exercises 1-22, evaluate the given expression. P(5,5)

Finite Mathematics for the Managerial, Life, and Social Sciences 