Chapter 17, Problem 15RE

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Chapter
Section

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

# Solve the boundary-value problem, if possible.15. y" + 4y' + 29y = 0, y(0) = 1, y(π) = −1

To determine

To solve: The boundary-value problem for differential equation y+4y+29y=0 , y(0)=1 , y(π)=1 .

Explanation

Given data:

The differential equation is,

yâ€³+4yâ€²+29y=0 (1)

Formula used:

Write the expression for a second order differential equation.

ayâ€³+byâ€²+cy=0 (2)

Write the expression for an auxiliary equation.

ar2+br+c=0 (3)

Write the expression for the complex roots.

r=Î±Â±iÎ² (4)

Here,

Î± is the real part of the root, and

Î² is the imaginary part of the root.

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

y(x)=eÎ±x(c1cosÎ²x+c2sinÎ²x) (5)

Write the expression to find the roots of quadratic equation.

r=âˆ’bÂ±b2âˆ’4ac2a (6)

Compare equation (1) and (2).

a=1b=4c=29

Substitute 1 for a, 4 for b, and 29 for c in equation (3),

(1)r2+(4)r+(29)=0r2+4r+29=0

Find the roots of auxiliary equation using equation (6).

Substitute 1 for a, 4 for b, and 29 for c in equation (6),

r=âˆ’(4)Â±(4)2âˆ’4(1)(29)2(1)=âˆ’4Â±16âˆ’1162=âˆ’4Â±1002=âˆ’4Â±i102

r=âˆ’2Â±i5 (7)

Compare equation (4) and (7)

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