   Chapter 17.1, Problem 25E

Chapter
Section
Textbook Problem

Solve the boundary-value problem, if possible.25. y" + 16y = 0, y(0) = –3, y(π/8) = 2

To determine

To solve: The boundary-value problem for differential equation y+16y=0 , y(0)=3 , y(π8)=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 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.

Write the required differential formulae to evaluate the differential equation.

ddxcosx=sinxddxsinx=cosx

Consider the differential equation as follows.

y+16y=0 (5)

Compare equation (1) and (5).

a=1b=0c=16

Find the auxiliary equation.

Substitute 1 for a , 0 for b and 16 for c in equation (2),

(1)r2+(0)r+(16)=0r2+16=0r2=16

Simplify equation as follows.

r=±i4 (6)

Compare equation (3) and (6).

α=0β=4

Find the general solution of y+16y=0 using equation (4).

Substitute 0 for α and 4 for β in equation (4),

y=e(0)x(c1cos4x+c2sin4x)

y=c1cos4x+c2sin4x (7)

Modify equation (7) as follows

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