Frequency Response
We are interested in the solution(s) to the linear constantcoefficient differential equa
tion
x
(
n
)
+
a
n

1
x
(
n

1)
+
· · ·
+
a
1
˙
x
+
a
0
x
=
b
m
u
(
m
)
+
· · ·
+
b
0
u
(1)
One approach is to use Laplace transforms and the resulting transfer function. Another
approach is based on the classical solution to the ordinary differential equation itself.
A particular solution
x
p
(
t
) is
any
solution that satisfies the differential equation. We
usually reserve the notation
x
p
(
t
), however, for a solution that does not contain any part
of the homogeneous, or complementary solution,
x
h
(
t
), obtained by solving the equation
with
u
≡
0.
Note that if
x
p
= sin
ωt
, then the lefthand side will be entirely in terms of sin
ωt
and
cos
ωt
. Similarly, if
x
p
is a polynomial or exponential in
t
, then the lefthand side will be
a polynomial or exponential in
t
.
So, if we have a certain form on the righthand side, we can try to get it by guessing
a related form for
x
h
.
For example, suppose we have the thirdorder ODE
x
(3)
+
a
2
¨
x
+
a
1
˙
x
+
a
0
x
=
K
sin
ωt
(2)
where the righthand side is the sinusoidal input used for frequency response analysis. We
guess a particular solution of the form
x
p
(
t
) =
A
sin
ωt
+
B
cos
ωt
(3)
where
A
and
B
are undetermined coefficients, but are different from the undetermined
coefficients that appear in the homogeneous solution. We differentiate the assumed so
lution three times and substitute the results into the original differential equation. That
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 Fall '08
 DEVENPORT
 Complex number, sin ωt

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