Chapter 17.1, Problem 28E

### 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.28. y" - 8y' + 17y = 0, y(0) = 3, y(π) = 2

To determine

To solve: The boundary-value problem for differential equation y8y+17y=0 , y(0)=3 , y(π)=2 .

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 two 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€³âˆ’8yâ€²+17y=0 (5)

Compare equation (1) and (5).

a=1b=âˆ’8c=17

Find the auxiliary equation.

Substitute 1 for a , âˆ’8 for b and 17 for c in equation (2),

(1)r2+(âˆ’8)r+(17)=0r2âˆ’8r+17=0

Solve for r .

r=âˆ’(âˆ’8)Â±(âˆ’8)2âˆ’4(1)(17)2(1)=8Â±64âˆ’682=8Â±âˆ’42=8Â±2i2

Simplify the equation as follows.

r=4Â±i (6)

Compare equation (3) and equation (6).

Î±=4Î²=1

Find the general solution of yâ€³âˆ’8yâ€²+17y=0 using equation (4).

Substitute 4 for Î± and 1 for Î² in equation (4),

y=e4x(c1cos(1)x+c2sin(1)x)

y=e4x(c1cosx+c2sinx) (7)

Modify equation (7) as follows

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