Page 171 - Numerical Analysis Using MATLAB and Excel

Chapter 5 Differential Equations, State Variables, and State Equations

If (5.28) is to be a real function of time, the constants k 1 and k 2 must be complex conjugates.
ϕ
The other constants k 3 , k 4 , k 5 , and the phase angle are real constants.
The forced response can be found by

a. The Method of Undetermined Coefficients or
b. The Method of Variation of Parameters

We will study the Method of Undetermined Coefficients first.

5.5 Using the Method of Undetermined Coefficients for the Forced Response

For simplicity, we will only consider ODEs of order 2 . Higher order ODEs are discussed in differ-
ential equations textbooks.
Consider the non-homogeneous ODE

2
d y d
a + b-----y + cy = fx() (5.29)
t d 2 dt
where , , and are real constants.ab c

We have learned that the total (complete) solution consists of the summation of the natural and
forced responses.

For the natural response, if y 1 and y 2 are any two solutions of (5.29), the linear combination
y = k y + k y , where k 1 and k 2 are arbitrary constants, is also a solution, that is, if we know
3
1 1
2 2
the two solutions, we can obtain the most general solution by forming the linear combination of
y 1 and y 2 . To be certain that there exist no other solutions, we examine the Wronskian Determi-
nant defined below.

y 1 y 2
d
d
Wy y ) ( 1 , 2 ≡ d d = y ------ y – y ------ y ≠ 0 (5.30)
1
2
1
2
------ y ------ y dx dx
dx 1 dx 2
WRONSKIAN DETERMINANT
If (5.30) is true, we can be assured that all solutions of (5.29) are indeed the linear combination of
y 1 and y 2 .
The forced response is obtained by observation of the right side of the given ODE as it is illus-
trated by the examples that follow.

5−10 Numerical Analysis Using MATLAB® and Excel®, Third Edition

Copyright © Orchard Publications