Mat 239 Differential Equations

Fall 2008

Laplace Transforms, Part 1 Due Wednesday, September 3rd

This is a project/assignment type thing. You can work with other people, but each person is required to turn in his or her own paper. All you'll need to do for this part is some calculations by hand, so just write everything up neatly like a homework assignment and turn it in, no need to type anything up, etc.

Introduction

The word transform, as it is usually used in math, is just a fancy word for a special type of function. (There are also things called operators which serve very similar roles.) Specifically, it is a function whose inputs are functions. You've actually seen a couple of these before, for example

EX The derivative operator, the simplest of which is, takes as an input an ordinary single variable function and spits out its derivative with respect to x. So if the input is the function , then.

EX In Calc 3, you probably defined a multivariable differential operator, . This guy could have several different types of "inputs," the simplest of which is a scalar function, . So, for example, if the input is, then .

The Laplace Transform is similarly an operator which takes as an input a function and spits out another function. Unlike the examples above, it's not directly related to differentiation, but its very useful nonetheless. Here's how the Laplace Transform is defined:

So the Laplace Transform takes a single variable function, puts it in the integrand of an improper integral along with an exponential term, and defines a new function which is those values of for which the improper integral converges. (Notationally, and indicate the exact same thing, and we go back and forth between the two forms as convenient.)

Let's look at an example.

EX Find the Laplace Transform of .

Given the function, we simply input it into the definition:

Now evaluate this improper integral:

Notice that the value of the integral is a function of . We now have to decide which values of will make this guy converge. We know that the exponential will have a limit of zero if its exponent is negative and will grow infinitely large if its exponent is positive. Thus, if , the integral will diverge, but if, it will converge and have a value of. Therefore the Laplace Transform of is for .

So we started with a normal function of , , and we ended up with a function of , .

It's probably not clear why we want instead of just, but the motivation will become clear later.

Assignment

I want you to find the Laplace Transforms for each of the following functions. Show all work involved in the integration and be sure to state the domain of the transform (, for example).

1.

2. (will require integration by parts and L'Hopital's rule)

3. (where is defined as)