There really isn’t all that much to this section. All we’re going to do here is work a quick
example using Laplace transforms for a 3rd order differential
equation so we can say that we worked at least one problem for a differential
equation whose order was larger than 2.
Everything that we know from the Laplace
Transforms chapter is still valid.
The only new bit that we’ll need here is the Laplace transform of the
third derivative. We can get this from
the general formula that we gave when we first
started looking at solving IVP’s with Laplace transforms. Here is that formula,
Here’s the example for this section.
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Example 1 Solve
the following IVP.
Solution
As always we first need to make sure the function
multiplied by the Heaviside function has been properly shifted.
It has been properly shifted and we can see that we’re
shifting . All we need to do now is take the Laplace
transform of everything, plug in the initial conditions and solve for . Doing all of this gives,
Now we need to partial fraction and inverse transform F(s) and G(s). We’ll leave it to
you to verify the details.
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.
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Okay, there is the one Laplace transform example with a
differential equation with order greater than 2. As you can see the work in identical except
for the fact that the partial fraction work (which we didn’t show here) is
liable to be messier and more complicated.