On occasion we will run across transforms of the form,
that can’t be dealt with
easily using partial fractions.
We would like a way to take the inverse transform of such a
transform. We can use a convolution
integral to do this.
Convolution Integral
A nice property of convolution integrals is.
Or,
The following fact will allow us to take the inverse
transforms of a product of transforms.
Fact
Let’s work a quick example to see how this can be used.
Example 1 Use
a convolution integral to find the inverse transform of the following
transform.
Solution
First note that we could use #11 from out table to do this one so that
will be a nice check against our work here.
Now, since we are going to use a convolution integral here
we will need to write it as a product whose terms are easy to find the
inverse transforms of. This is easy to
do in this case.
So, in this case we have,
Using a convolution integral h(t) is,
This is exactly what we would have gotten by using #11
from the table.

Convolution integrals are very useful in the following kinds
of problems.
Example 2 Solve
the following IVP
Solution
First, notice that the forcing function in this case has
not been specified. Prior to this
section we would not have been able to get a solution to this IVP. With convolution integrals we will be able
to get a solution to this kind of IVP.
The solution will be in terms of g(t)
but it will be a solution.
Take the Laplace
transform of all the terms and plug in the initial conditions.
Notice here that all we could do for the forcing function
was to write down G(s) for its
transform. Now, solve for Y(s).
We factored out a 4 from the denominator in preparation
for the inverse transform process. To
take inverse transforms we’ll need to split up the first term and we’ll also
rewrite the second term a little.
Now, the first two terms are easy to inverse
transform. We’ll need to use a
convolution integral on the last term.
The two functions that we will be using are,
We can shift either of the two functions in the
convolution integral. We’ll shift g(t) in our solution. Taking the inverse transform gives us,
So, once we decide on a g(t) all we need to do is to an integral and we’ll have the
solution.

As this last example has shown, using convolution integrals
will allow us to solve IVP’s with general forcing functions. This could be very convenient in cases where
we have a variety of possible forcing functions and don’t know which one we’re going
to use. With a convolution integral all
that we need to do in these cases is solve the IVP once then go back and
evaluate an integral for each possible g(t). This will save us the work of having to solve
the IVP for each and every g(t).