   Chapter 17, Problem 16RE

Chapter
Section
Textbook Problem

Solve the boundary-value problem, if possible.16. y" + 4y' + 29y = 0, y(0) = 1, y(π) = −e-2π

To determine

To solve: The boundary-value problem for differential equation y+4y+29y=0 , y(0)=1 , y(π)=e2π .

Explanation

Given data:

The differential equation is,

y+4y+29y=0 (1)

Formula used:

Write the expression for differential equation.

ay+by+cy=0 (2)

Write the expression for auxiliary equation.

ar2+br+c=0 (3)

Write the expression for the complex roots.

r=α±iβ (4)

Here,

α is the real part of the root, and

β is the imaginary part of the root.

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

y(x)=eαx(c1cosβx+c2sinβx) (5)

Write the expression to find the roots of quadratic equation.

r=b±b24ac2a (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=(4)±(4)24(1)(29)2(1)=4±161162=4±1002=4±i102

r=2±i5 (7)

Compare equation (4) and (7)

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