Chapter 17.1, Problem 12E

### 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.12.   d 2 R d t 2 +   6 d R d t   +   34 R   =   0

To determine

To solve: The differential equation of d2Rdt2+6dRdt+34R=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.

Write the expression to find the roots of quadratic equation.

r=âˆ’bÂ±b2âˆ’4ac2a (5)

Consider the differential equation as follows.

d2Rdt2+6dRdt+34R=0 (6)

Modify equation (1) as follows.

Compare equation (6) and (7).

a=1b=6c=34

Find the auxiliary equation.

Substitute 1 for a , 6 for b and 34 for c in equation (2),

(1)r2+(6)r+(34)=0r2+6r+34=0

Find the roots of equation using equation (5)

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