Page 181 - Numerical Analysis Using MATLAB and Excel

Chapter 5 Differential Equations, State Variables, and State Equations

5.6 Using the Method of Variation of Parameters for the Forced Response

In certain non−homogeneous ODEs, the right side ft() cannot be determined by the method of
undetermined coefficients. For these ODEs we must use the method of variation of parameters.
This method will work with all linear equations including those with variable coefficients such as

d y dy
2
-------- + α t()------ + β t()y = ft() (5.67)
dt 2 dt
provided that the general form of the natural response is known.

Our discussion will be restricted to second order ODEs with constant coefficients.

The method of variation of parameters replaces the constants k 1 and k 2 by two variables u 1 and
u 2 that satisfy the following three relations:

y = u y + u y (5.68)
1 1
2 2
du du
2
1
------- y + -------- y = 0 (5.69)
dt 1 dt 2
du dy du dy
2
1
1
2
⋅
⋅
-------- -------- + -------- -------- = ft() (5.70)
dt dt dt dt
⁄
Simultaneous solution of (5.68) and (5.69) will yield the values of du dt and du ⁄ dt ; then, inte-
2
1
gration of these will produce u 1 and u 2 , which when substituted into (5.67) will yield the total
solution.
Example 5.12
Find the total solution of
dy
2
d y 4------ + 3y = 12 (5.71)
-------- +
dt 2 dt
in terms of the constants k 1 and k 2 by the

a. method of undetermined coefficients

b. method of variation of parameters
Solution:

With either method, we must first find the natural response. The characteristic equation yields

5−20 Numerical Analysis Using MATLAB® and Excel®, Third Edition

Copyright © Orchard Publications