Chapter 17, Problem 11RE

### 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.11. y" + 6y' = 0, y(1) = 3, y'(1) = 12

To determine

To solve: The initial-value problem for differential equation y+6y=0 , y(1)=3 , y(1)=12 .

Explanation

Given data:

The differential equation is,

yâ€³+6yâ€²=0 (1)

Formula used:

Write the expression for differential equation.

ayâ€³+byâ€²+cy=0 (2)

Write the expression for auxiliary equation.

ar2+br+c=0 (3)

Write the expression for general solution of ayâ€³+byâ€²+cy=0 with two distinct real roots r .

y(x)=c1er1x+c2er2x (4)

Here,

r1 and r2 are the root of auxiliary equation.

Compare equation (1) and (2).

a=1b=6c=0

Substitute 1 for a , 6 for b and 0 for c in equation (3),

(1)r2+(6)r+(0)=0r2+6r=0r(r+6)=0

Simplify equation as follows.

r=0r+6=0r=âˆ’6

Consider the value of r1 and r2 as follows.

r1=0r2=âˆ’6

Find the general solution of yâ€³+6yâ€²=0 using equation (4).

Substitute 0 for r1 and âˆ’6 for r2 in equation (4),

y(x)=c1e(0)x+c2e(âˆ’6)x

y(x)=c1+c2eâˆ’6x (5)

Modify equation (5) as follows

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