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Discontinuous RHS
The method of Laplace transforms gives us a way of dealing with the nonhomogeneous equation
y
00
+
p
(
x
)
y
0
+
q
(
x
)
y
=
g
(
x
)
when
g
(
x
)
is piecewise continuous, which for our purposes we will take to mean a function that is
continuous except for ﬁnitely many jump discontinuities. We will construct our piecewise continuous
functions in terms of the following important function:
Definition
The piecewise continuous function
H
c
(
x
)
is deﬁned by the formula
H
c
(
x
) =
±
0
0
≤
x < c
1
x
≥
c
This is called a Heaviside function after Oliver Heaviside.
Now we can write piecewise continuous functions in terms of the Heaviside function as demonstrated
in the following example.
I
Example
Rewrite the piecewise continuous function
g
(
x
) =
x
2
0
≤
x <
3
x
3
≤
x <
6
5
x
≥
6
in terms of Heaviside functions.
Solution
First we will construct functions that are 1 on each interval of the intervals of deﬁnition
appearing in
g
(
x
)
and 0 oﬀ of these intervals.
The function
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