Step functions
April 9, 2011
Recall the Heaviside step function (or unitstep function) is deﬁned as
u
c
(
t
) =
(
0
x < c
1
x > c
(1)
Also recall the “tshifting” theorem (as opposed to the “sshifting” theorem):
L
{
u
c
(
t
)
f
(
t

c
)
}
=
e

cs
L
{
f
(
t
)
}
.
(2)
In order to solve diﬀerential equations with discontinuous forcing functions, we express them
in terms of the Heaviside functions, and use above theorem to take their Laplace transforms.
Problem statement:
(“Forward” problem) Find the Laplace transform of
g
(
t
) =
u
c
(
t
)
h
(
t
)
(3)
Steps:
1. Identify
f
(
t

c
)
. This function is what happens to be multiplied
by
u
c
(
t
)
. In problem statement above,
f
(
t

c
) =
h
(
t
)
.
2. Evaluate
f
(
t
)
from Step 1. This means replacing
t
with
t
+
c
in
the identiﬁed function.
3. From Eq. (
2
),
L
{
g
(
t
)
}
=
e

cs
L
{
f
(
t
)
}
, where
f
(
t
)
is the func
tion determined in Step 2.
Problem statement:
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '10
 None
 Laplace

Click to edit the document details