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Laplace Transform Properties
By building up some basic properties of the Laplace transform, we can expand the list of functions
we know the transform of, thus increasing the number of IVP’s we can solve by this method.
Property 1:
If
F
(
s
) =
L
[
f
(
x
)]
, then
L
[

xf
(
x
)] =
F
0
(
s
)
.
In words, multiplying by

x
in our usual function space is the same as diﬀerentiation in transform
space.
I
Example
Find a function whose Laplace transform is
1
(
s

a
)
2
.
Solution
We have that
d
ds
±

1
s

a
²
=
1
(
s

a
)
2
Furthermore, we know that
L
[

e
ax
] =

1
s

a
so by Property 1 we have
L
[
xe
ax
] =
L
[

x
(

e
ax
)] =
d
ds
±

1
s

a
²
=
1
(
s

a
)
2
.
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View Full Document Property 2:
If
F
(
s
) =
L
[
f
(
x
)]
, then
L
[
e
ax
f
(
x
)] =
F
(
s

a
)
.
In words, multiplying by
e
ax
in our usual function space is the same as translation to the right by
a
in transform space.
I
Example
Find a function whose Laplace transform is
1
s
2

4
s
+ 9
.
Solution
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This note was uploaded on 03/03/2012 for the course MATH 4430 at Colorado.
 '08
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