In this section we will use Laplace
transforms to solve IVP’s which contain Heaviside functions in the forcing
function. This is where Laplace
transform really starts to come into its own as a solution method.
To work these problems we’ll just need to remember the
following two formulas,
In other words, we will always need to remember that in
order to take the transform of a function that involves a Heaviside we’ve got
to make sure the function has been properly shifted.
Let’s work an example.
Example 1 Solve
the following IVP.
Solution
First let’s rewrite the forcing function to make sure that
it’s being shifted correctly and to identify the function that is actually
being shifted.
So, it is being shifted correctly and the function that is
being shifted is . Taking the Laplace
transform of everything and plugging in the initial conditions gives,
Now solve for Y(s).
Notice that we combined a couple of terms to simplify
things a little. Now we need to
partial fraction F(s) and G(s).
We’ll leave it to you to check the details of the partial fractions.
We now need to do the inverse transforms on each of
these. We’ll start with F(s).
Now G(s).
Okay, we can now get the solution to the differential
equation. Starting with the transform
we get,
where f(t) and g(t) are the functions shown above.

There is can be a fair amount of work involved in solving
differential equations that involve Heaviside functions.
Let’s take a look at another example or two.
Example 2 Solve
the following IVP.
Solution
Let’s rewrite the differential equation so we can identify
the function that is actually being shifted.
So, the function that is being shifted is and it is being shifted correctly. Taking the Laplace
transform of everything and plugging in the initial conditions gives,
Now solve for Y(s).
Notice that we combined the first two terms to simplify
things a little. Also there was some
canceling going on in this one. Do not
expect that to happen on a regular basis.
We now need to partial fraction F(s). We’ll leave the details to you to check.
Okay, we can now get the solution to the differential
equation. Starting with the transform
we get,
where f(t) is
given above.

Example 3 Solve
the following IVP.
Solution
Let’s take the Laplace
transform of everything and note that in the third term we are shifting 4t.
Now solve for Y(s).
So, we have three functions that we’ll need to partial
fraction for this problem. I’ll leave
it to you to check the details.
Okay, we can now get the solution to the differential
equation. Starting with the transform
we get,
where f(t), g(t)
and h(t) are given above.

Let’s work one more example.
Example 4 Solve
the following IVP.
where,
Solution
The first step is to get g(t) written in terms of Heaviside functions so that we can take
the transform.
Now, while this is g(t)
written in terms of Heaviside functions it is not yet in proper form for us
to take the transform. Remember that
each function must be shifted by a proper amount. So, getting things set up for the proper
shifts gives us,
So, for the first Heaviside it looks like is the function that is being shifted and
for the second Heaviside it looks like is being shifted.
Now take the Laplace
transform of everything and plug in the initial conditions.
Solve for Y(s).
Now, in the solving process we simplified things into as
few terms as possible. Even doing
this, it looks like we’ll still need to do three partial fractions.
I’ll leave the details of the partial fractioning to you
to verify. The partial fraction form
and inverse transform of each of these are.
Putting this all back together is going to be a little
messy. First rewrite the transform a
little to make the inverse transform process possible.
Now, taking the inverse transform of all the pieces gives
us the final solution to the IVP.
where f(t), g(t), and h(t) are defined above.

So, the answer to this example is a little messy to write
down, but overall the work here wasn’t too terribly bad.
Before proceeding with the next section let’s see how we
would have had to solve this IVP if we hadn’t had Laplace
transforms. To solve this IVP we would
have had to solve three separate IVP’s.
One for each portion of g(t). Here is a list of the IVP’s that we would
have had to solve.
 0 < t < 6
The solution
to this IVP, with some work, can be made to look like,

where, y_{1}(t) is the solution to the first IVP. The solution to this IVP, with some work, can
be made to look like,

where, y_{2}(t) is the solution to the second IVP. The solution to this IVP, with some work, can
be made to look like,
There is a considerable amount of work required to solve all
three of these and in each of these the forcing function is not that
complicated. Using Laplace
transforms saved us a fair amount of work.