   Chapter 17.1, Problem 28E

Chapter
Section
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.

ay+by+cy=0 (1)

Write the expression for auxiliary equation.

ar2+br+c=0 (2)

Write the expression for the complex roots.

r=α±iβ (3)

Write the expression for general solution of ay+by+cy=0 with two complex roots.

y=eαx(c1cosβx+c2sinβx) (4)

Here,

α is the real part of the root, and

β is the imaginary part of the root.

Consider the differential equation as follows.

y8y+17y=0 (5)

Compare equation (1) and (5).

a=1b=8c=17

Find the auxiliary equation.

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

(1)r2+(8)r+(17)=0r28r+17=0

Solve for r .

r=(8)±(8)24(1)(17)2(1)=8±64682=8±42=8±2i2

Simplify the equation as follows.

r=4±i (6)

Compare equation (3) and equation (6).

α=4β=1

Find the general solution of y8y+17y=0 using equation (4).

Substitute 4 for α and 1 for β 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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