This post is an introduction to the Laplace transform and how it can be used to solve differential equations. Let be a function. We say that is **locally integrable** if for any compact ,

(Recall that a subset of is compact if and only if it is closed and bounded.) For a locally integrable function , we define the **Laplace transform** of by

for any complex for which the above integral exists. (For the rest of this post, assume that all functions are defined on .) For the most basic example, let . Then

We can generalize the above with the following theorem:

Theorem 1For any nonnegative integer ,

*Proof:* We proved the base case above. Now suppose the claim holds for some . Then

Using integration by parts, the above is equal to

and further computation yields

Now suppose is an infinitely differentiable, locally integrable function on . Then we have the following theorem:

Theorem 2For any nonnegative integer ,

where and is the derivative of , with .

*Proof:* For , the claim is evident:

Assume the claims holds for some nonnegative integer , then we compute

Another example; suppose and for . Then

for .

One more theorem:

Theorem 3The Laplace transform is a linear operator. That is, if and are locally integrable functions, and , thenand

*Proof:* This follows from linearity of the Lebesgue integral. Left as an exercise to the reader 🙂

Now for an example, suppose is a locally integrable, twice-differentiable function defined on satisfying the differential equation

and further that

Applying the Laplace transform to the LHS, we have (letting )

As , we have

and hence

Since

it follows that