Digital Signal Processing Reference
In-Depth Information
R
i
(
t
)
v
(
t
)
L
Figure 3.17
RL circuit.
A simple example of an ordinary linear differential equation with constant
coefficients is the first-order differential equation
dy(t)
dt
+ 2y(t) = x(t).
(3.49)
This equation could model the circuit of Figure 3.17, namely,
L
di(t)
dt
+ Ri(t) = v(t),
with and
The general form of an
n
th-order linear differential equation with constant
coefficients is
L = 1H, R = 2Æ, y(t) = i(t),
x(t) = v(t).
d
n
y(t)
dt
n
d
n- 1
y(t)
dt
n- 1
dy(t)
dt
+
Á
+ a
1
a
n
+ a
n- 1
+ a
0
y(t)
d
m
x(t)
dt
m
d
m- 1
x(t)
dt
m- 1
dx(t)
dt
+
Á
+ b
1
= b
m
+ b
m- 1
+ b
0
x(t),
where and are constants and We limit these constants
to having real values. This equation can be expressed in the more compact form
a
0
,
Á , a
n
b
0
,
Á , b
m
a
n
Z 0.
d
k
y(t)
dt
k
d
k
x(t)
dt
k
n
m
a
k
=
a
b
k
.
(3.50)
a
k= 0
k= 0
It is easily shown by the preceding procedure that this equation is both linear and
time invariant. Many methods of solution exist for (3.50); in this section, we review
briefly one of the classical methods. In subsequent chapters, the solution by trans-
form methods is developed.
The method of solution of (3.50) presented here is called the
method of undeter-
mined coefficients
[2] and requires that the general solution
y(t)
be expressed as the
sum of two functions:
y(t) = y
c
(t) + y
p
(t).
(3.51)
