   # Solve the initial-value problem. 9 y ″ + 12 y ′ + 4 y = 0 , y ( 0 ) = 1 , y ′ ( 0 ) = 0 ### Calculus (MindTap Course List)

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781285740621

#### Solutions

Chapter
Section ### Calculus (MindTap Course List)

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781285740621
Chapter 17.1, Problem 19E
Textbook Problem
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## Solve the initial-value problem. 9 y ″ + 12 y ′ + 4 y = 0 ,    y ( 0 ) = 1 ,    y ′ ( 0 ) = 0

To determine

To solve:

The initial value problem 9y''+12y'+4y=0  ,  y0=1,  y'0=0

Solution:

y=e-23x+23xe-23x

Explanation:

1) Concept:

i) Theorem: If y1 and y2 are linearly independent solutions of the second order Homogeneous linear differential equation ay''+by'+cy=0 on an interval, and   a0, then the general solution is given by yx=c1y1x+c2y2(x) where c1 and c2 are constants.

ii) In general solution of ay''+by'+cy=0

 Roots of ar2+br+c=0 General solution r1, r2  are real and distinct y=c1er1x+c2er2x r1=r2=r y=c1erx+c2xerx r1 , r2 complex :α±iβ y=eαx(c1cosβx+c2sinβx)

2) Given:

The initial value problem 9y''+12y'+4y=0  ,  y0=1,  y'0=0

3) Calculations:

The auxiliary equation of the differential equation  9y''+12y'+4y=0  is  9r2+12r+4=0

Solving the auxiliary equation   9r2+12r+4=0

9r2+12r+4=0

9r2+6r+6r+4=0

3r(3r+2)+2(3r+2)=0

(3r+2)(3r+2)=0

r=-23,-23

By given concept general solution of differential equation 9y''+12y'+4y=0   is

y=c1e-23x+c2xe-23x

Differentiate this equation with respect to x

y=-23c1e-23x+c2e-23x-23c2xe-23x

Applying initial conditions y0=1,  y'0=0

y0=c1  Therefore ,  c1=1

y'0=-23c1+c2  Therefore,  -23c1+c2=0 this imply c2=23

Thus, the solution of the initial value problem y''-2y'-3y=0  , y0=1,  y'0=0  is y=e-23x+23xe-23x

Conclusion:

The solution of the initial value problem y''-2y'-3y=0  , y0=2,  y'0=2   is y=e-23x+23xe-23x

...

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