   Chapter 17.1, Problem 32E

Chapter
Section
Textbook Problem

Solve the boundary-value problem, if possible.32. y" + 4y' + 20y = 0, y(0) = 1, y(π) = e-2π

To determine

To solve: The boundary-value problem for differential equation y+4y+20y=0 , y(0)=1 , y(π)=e2π .

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 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.

y+4y+20y=0 (5)

Compare equation (1) and (5).

a=1b=4c=20

Find the auxiliary equation.

Substitute 1 for a , 4 for b and 20 for c in equation (2),

(1)r2+(4)r+(20)=0r2+4r+20=0

Solve for r .

r=(4)±(4)24(1)(20)2(1)=4±642=4±8i2

Simplify the equation as follows.

r=2±i4 (6)

Compare equation (3) and (6).

α=2β=4

Find the general solution of y+4y+20y=0 using equation (4).

Substitute 2 for α and 4 for β in equation (4),

y=e2x(c1cos(4)x+c2sin(4)x)

y=e2x(c1cos4x+c2sin4x) (7)

Modify equation (7) as follows.

y(x)=e2x(c1cos4x+c2sin4x) (8)

Find the value of y(0)

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