Response of the 1
st
Order System to an Arbitrary Input
A 1
st
order system is described by an ordinary diﬀerential equation (ODE),
a
1
dy
(
t
)
dt
+
a
0
y
(
t
) =
x
(
t
)
,
(1)
where
a
0
and
a
1
are constants. Input
x
(
t
) is a known function, such that
x
(
t
) = 0
for
t <
0
.
(2)
We want to ﬁnd the solution of this equation subject to an initial condition (IC),
y
(0

) = 0
.
(3)
We ﬁrst divide both sides of (1) by
a
0
:
a
1
a
0
dy
(
t
)
dt
+
y
(
t
) =
1
a
0
x
(
t
)
(4)
and deﬁne
τ
=
a
1
/a
0
(time constant). Hence, equation to be solved takes form:
τ
dy
(
t
)
dt
+
y
(
t
) =
1
a
0
x
(
t
)
(5)
Finding the solution requires two steps: solving a homogeneous equation, and ﬁnding solution
to the nonhomogeneous equation (5) by variation of parameter.
Solution to Homogeneous Equation
Homogeneous equation is obtained from (5) by assuming
x
(
t
) = 0 for all
t
:
τ
dy
(
t
)
dt
+
y
(
t
) = 0
.
(6)
We rearrange the terms so that dependent variable
y
appears only on the lefthand side and
independent variable
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 Spring '08
 IZATT
 Calculus, Derivative, homogeneous equation

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