Chapter 17.1, Problem 3E

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Chapter
Section

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

# Solve the differential equation.3. y" + 2y = 0

To determine

To solve: The differential equation of y+2y=0 .

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 the complex roots.

r=Î±Â±iÎ² (3)

Write the expression for general solution of ayâ€³+byâ€²+cy=0 with complex roots.

y=eÎ±x(c1cosÎ²x+c2sinÎ²x) (4)

Here,

Î± is the real part of the root, and

Î² is the imaginary part of the root.

Consider the differential equation as follows.

yâ€³+2y=0 (5)

Compare equation (1) and (5).

a=1b=0c=2

Find the auxiliary equation.

Substitute 1 for a , 0 for b and 2 for c in equation (2),

(1)r2+(0)r+(2)=0r2+2=0

Re-write equation as follows.

r2=âˆ’2r=Â±2i

Write the expression of r in terms of Î± and Î²

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