Concept explainers
To solve:
The initial value problem 9 y ' ' + 12 y ' + 4 y = 0 , y 0 = 1 , y ' 0 = 0
Solution:
y = e - 2 3 x + 2 3 x e - 2 3 x
Explanation:
1) Concept:
i) Theorem: If y 1 and y 2 are linearly independent solutions of the second order Homogeneous linear differential equation a y ' ' + b y ' + c y = 0 on an interval, and a ≠ 0 , then the general solution is given by y x = c 1 y 1 x + c 2 y 2 ( x ) where c 1 a n d c 2 are constants.
ii) In general solution of a y ' ' + b y ' + c y = 0
Roots of a r 2 + b r + c = 0
General solution
r 1 , r 2 are real and distinct
y = c 1 e r 1 x + c 2 e r 2 x
r 1 = r 2 = r
y = c 1 e r x + c 2 x e r x
r 1 , r 2 c o m p l e x : α ± i β
y = e α x ( c 1 c o s β x + c 2 s i n β x )
2) Given:
The initial value problem 9 y ' ' + 12 y ' + 4 y = 0 , y 0 = 1 , y ' 0 = 0
3) Calculations:
The auxiliary equation of the differential equation 9 y ' ' + 12 y ' + 4 y = 0 is 9 r 2 + 12 r + 4 = 0
Solving the auxiliary equation 9 r 2 + 12 r + 4 = 0
9 r 2 + 12 r + 4 = 0
9 r 2 + 6 r + 6 r + 4 = 0
3 r ( 3 r + 2 ) + 2 ( 3 r + 2 ) = 0
( 3 r + 2 ) ( 3 r + 2 ) = 0
r = - 2 3 , - 2 3
By given concept general solution of differential equation 9 y ' ' + 12 y ' + 4 y = 0 is
y = c 1 e - 2 3 x + c 2 x e - 2 3 x
Differentiate this equation with respect to x
y = - 2 3 c 1 e - 2 3 x + c 2 e - 2 3 x - 2 3 c 2 x e - 2 3 x
Applying initial conditions y 0 = 1 , y ' 0 = 0
y 0 = c 1 Therefore , c 1 = 1
y ' 0 = - 2 3 c 1 + c 2 Therefore, - 2 3 c 1 + c 2 = 0 this imply c 2 = 2 3
Thus, the solution of the initial value problem y ' ' - 2 y ' - 3 y = 0 , y 0 = 1 , y ' 0 = 0 is y = e - 2 3 x + 2 3 x e - 2 3 x
Conclusion:
The solution of the initial value problem y ' ' - 2 y ' - 3 y = 0 , y 0 = 2 , y ' 0 = 2 is y = e - 2 3 x + 2 3 x e - 2 3 x
To solve:
The initial value problem
Solution:
Explanation:
1) Concept:
i) Theorem: If
ii) In general solution of
Roots of |
General solution |
2) Given:
The initial value problem
3) Calculations:
The auxiliary equation of the differential equation
Solving the auxiliary equation
By given concept general solution of differential equation
Differentiate this equation with respect to
Applying initial conditions
Thus, the solution of the initial value problem
Conclusion:
The solution of the initial value problem
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Chapter 17 Solutions
Calculus (MindTap Course List)
- Calculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,