In this chapter we will be looking at how to use Laplace transforms to solve differential equations. There are many kinds of transforms out there
in the world. Laplace
transforms and Fourier transforms are probably the main two kinds of transforms
that are used. As we will see in later
sections we can use Laplace transforms to
reduce a differential equation to an algebra problem. The algebra can be messy on occasion, but it
will be simpler than actually solving the differential equation directly in
many cases. Laplace
transforms can also be used to solve IVP’s that we can’t use any previous
method on.
For “simple” differential equations such as those in the
first few sections of the last chapter Laplace transforms will be more
complicated than we need. In fact, for
most homogeneous differential equations such as those in the last chapter Laplace transforms is significantly longer and not so
useful. Also, many of the “simple”
nonhomogeneous differential equations that we saw in the Undetermined Coefficients and Variation of Parameters are still simpler
(or at the least no more difficult than Laplace transforms) to do as we did
them there. However, at this point, the
amount of work required for Laplace transforms
is starting to equal the amount of work we did in those sections.
Laplace transforms comes
into its own when the forcing function in the differential equation starts
getting more complicated. In the
previous chapter we looked only at nonhomogeneous differential equations in
which g(t) was a fairly simple
continuous function. In this chapter we
will start looking at g(t)’s that are
not continuous. It is these problems
where the reasons for using Laplace transforms
start to become clear.
We will also see that, for some of the more complicated
nonhomogeneous differential equations from the last chapter, Laplace
transforms are actually easier on those problems as well.
Here is a brief rundown of the sections in this chapter.
The Definition 
The definition of the Laplace
transform. We will also compute a couple
Laplace transforms using the definition.
Laplace Transforms As the previous section will demonstrate,
computing Laplace transforms directly from the
definition can be a fairly painful process.
In this section we introduce the way we usually compute Laplace transforms.
Inverse Laplace Transforms In this section we ask the opposite
question. Here’s a Laplace
transform, what function did we originally have?
Step Functions This is one of the more important functions in
the use of Laplace transforms. With the introduction of this function the
reason for doing Laplace transforms starts to
become apparent.
Solving IVP’s with Laplace Transforms
Here’s how we used Laplace
transforms to solve IVP’s.
Nonconstant
Coefficient IVP’s We will see how Laplace
transforms can be used to solve some nonconstant coefficient IVP’s
IVP’s with Step Functions Solving IVP’s that contain step
functions. This is the section where the
reason for using Laplace transforms really
becomes apparent.
Dirac Delta Function One last function that often shows up in Laplace transform problems.
Convolution Integral A brief introduction to the convolution
integral and an application for Laplace
transforms.
Table of Laplace Transforms This is a small table of Laplace Transforms
that we’ll be using here.