The Laplace Transform
Among the tools are useful for solving differential linear equations are integral
transforms.
Definition
: An
integral transform
is a relation of the form
F(s)
=
K(s,t)f(t)dt
a
b
!
where K(s,t) is a given function of the variables s and t, called the
kernel
of the
transformation and the limits of integration a and b are given and it is possible that a = 
∞
and b =
∞
.
The relation transforms the function f(t) into another function F(s), which is called the
transform
of f.
The general idea of using an integral transform to solve a linear differential equation is to
transform a problem for an unknown function f into a simpler problem for the function F.
Then, solve the simpler problem to find F and then recover the function f from its
transform F. This last step is known as inverting the transform.
There are several integral transform that are useful to solve differential equation, but in
this chapter we will use the Laplace transform.
Definition
: Let f(t) be given for t
≥
0 and assume the function satisfy certain conditions
to be stated later on. The
Laplace transform
of f(t), that it is denoted by
L
{f(t)} or F(s),
is defined by the equation
L
{f(t)} = F(s) =
e
!
st
f(t)dt
0
"
#
whenever the improper integral converges.
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 Fall '08
 STAFF
 Derivative, Mathematical analysis, dt

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