Page 204 - Numerical Analysis Using MATLAB and Excel

Summary

yt() = y Natural + y Forced = y + y F
N
Response Response
The natural response y N contains arbitrary constants and these can be evaluated from the

given initial conditions. The forced response y F , however, contains no arbitrary constants. It is
imperative to remember that the arbitrary constants of the natural response must be evaluated
from the total response.

• For an nth order homogeneous differential equation the solutions are
s t s t s t s t
y = k e , 1 1 y = k e , 2 2 y = k e , 3 3 … , y = k e n
1
n
3
2
n
,
,
,
where s s … s n are the solutions of the characteristic equation
1
2
n
a s + a n – 1 s n – 1 + … + a s + a 0 = 0
n
1
,
and a a, n n – 1 , … a a 0 are the constant coefficients of the ODE
,
1
• If the roots of the characteristic equation are distinct, the solutions of the natural response
n
are independent and the most general solution is:
s t s t s t
y = k e 1 + k e 2 + … + k e n
1
2
N
n
• If the solution of the characteristic equation contains equal roots, the most general solution
m
has the form:
s t s t s t
1
y = ( k + k t + … + k t m – 1 ) e + k n – i e 2 + … + k e n
2
m
N
1
n
• If the characteristic equation contains complex roots, these occur as complex conjugate pairs.
β
α
Thus, if one root is s = – α + jβ where and are real numbers, then another root is
1
s = – α jβ . Then, for two complex conjugate roots we evaluate the constants from the
–
2
expressions
s t s t – αt – αt
1
)
2
k e + k e = e ( k cos βt + k sin βt = e k cos ( βt + ϕ )
5
1
2
3
4
• The forced response of a non−homogeneous ODE can be found by the method of undeter-
mined coefficients or the method of variation of parameters.
• With the method of undetermined coefficients, the forced response is a function similar to the
right side of the non−homogeneous ODE. The form of the forced response for second order
non−homogeneous ODEs is given in Table 5.1.
• 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 parame-
Numerical Analysis Using MATLAB® and Excel®, Third Edition 5−43
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