   Chapter 17.1, Problem 17E

Chapter
Section
Textbook Problem

Solve the initial-value problem.17. y" + 3y = 0, y(0) = 1, y'(0) = 3

To determine

To solve: The initial-value problem for differential equation y+3y=0 , y(0)=1 , y(0)=3 .

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+3y=0 (5)

Compare equation (1) and (5).

a=1b=0c=3

Find the auxiliary equation.

Substitute 1 for a , 0 for b and 3 for c in equation (2),

(1)r2+(0)r+(3)=0r2+3=0

Re-write equation as follows.

r2=3r=±3i

Write the expression of r in terms of α and β .

r=0±i3 (6)

Compare equation (3) and (6).

α=0β=3

Find the general solution of y+3y=0 using equation (4).

Substitute 0 for α and 3 for β in equation (4),

y=e(0)x(c1cos3x+c2sin3x)=(1)(c1cos3x+c2sin3x)

y=c1cos3x+c2sin3x (7)

Modify equation (7) as follows.

y(x)=c1cos3x+c2sin3x (8)

Find the value of y(0)

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