Chapter 17.1, Problem 20E

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Chapter
Section

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
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.

ayâ€³+byâ€²+cy=0 (1)

Write the expression for auxiliary equation.

ar2+br+c=0 (2)

Write the expression for general solution of ayâ€³+byâ€²+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.

3yâ€³âˆ’2yâ€²âˆ’y=0 (4)

Compare equation (1) and (4).

a=3b=âˆ’2c=âˆ’1

Find the auxiliary equation.

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

(3)r2+(âˆ’2)r+(âˆ’1)=03r2âˆ’2râˆ’1=0(3r+1)(râˆ’1)=0

Simplify the equation as follows.

3r+1=0r=âˆ’13râˆ’1=0r=1

Consider the value of r1 and r2 as follows.

r1=âˆ’13r2=1

Find the general solution of 3yâ€³âˆ’2yâ€²âˆ’y=0 using equation (3).

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

y=c1eâˆ’13x+c2ex

y=c1eâˆ’x3+c2ex (5)

Modify equation (5) as follows.

y(x)=c1eâˆ’x3+c2ex (6)

Find the value of y(0)

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