   Chapter 17.1, Problem 19E

Chapter
Section
Textbook Problem

Solve the initial-value problem.19. 9y" + 12y' + 4y = 0, y(0) = 1, y'(0) = 0

To determine

To solve: The initial-value problem for differential equation 9y+12y+4y=0 , y(0)=1 , y(0)=0 .

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 general solution of ay+by+cy=0 with same real roots.

y=c1erx+c2xerx (3)

Here,

r is the root of auxiliary equation.

Write the required differential formulae to evaluate the differential equation.

ddxenx=nenx

Consider the differential equation as follows.

9y+12y+4y=0 (4)

Compare equation (1) and (4).

a=9b=12c=4

Find the auxiliary equation.

Substitute 9 for a , 12 for b and 4 for c in equation (2),

(9)r2+(12)r+(4)=09r2+12r+3=0(3r+2)2=0(3r+2)=0,(3r+2)=0

Simplify the equation as follows.

3r=2,3r=2r=23,23

Find the general solution of 9y+12y+4y=0 using equation (3).

Substitute 23 for r in equation (3),

y=c1e23x+c2xe23x (5)

Modify equation (5) as follows.

y(x)=c1e23x+c2xe23x (6)

Find the value of y(0) .

Substitute 0 for x in equation (6),

y(0)=c1e23(0)+c2(0)e23(0)=c1e0+0=c1

Substitute 1 for y(0) ,

1=c1

Differentiate equation (6) with respect to x

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