But what is a Laplace Transform?

What you're looking at, a somewhat complicated diagram that you and I are going to build up in this video, is a visualization unpacking the meaning behind one of the most powerful tools used to study differential equations. It's known as the lelass transform. This is one of those tools where as a student, you can learn how to use it to solve equations and yet be left completely in the dark about what it's actually doing. Think about learning how to drive a car versus learning how an internal combustion engine works.

Both are worthy pursuits. One is not necessarily better than the other, and in fact, driving is probably more practical. But there is something deeply satisfying about popping open the hood and understanding the mechanism inside. Similarly, our main goal with this video is to pop the hood and show you some of the beautiful math that awaits us inside this object.

And strictly speaking, a lot of what I want to show is not necessary if your only goal is to solve equations. That said, for the differential equation students among you, I think the content here should make the stuff you have to memorize a lot more memorable. And after dissecting this machine, studying it piece by piece, you and I will take everything for a test drive and see what it looks like to solve a concrete and very interesting differential equation. Now before I just plop down the definition on the screen, let's talk about what problem the applause transform is trying to solve.

We set up a lot of this in the previous chapter. And there are two primary ideas worth restating here. Number one, you need to understand exponential functions. And I'm always going to be writing these as e to the s * t.

t represents time. And then s is a number. It determines what specific exponential we're talking about. But very importantly for this topic, we're going to give S the freedom to take on complex number values.

We covered this much more thoroughly in the previous chapter, but the quick summary is that if S has an imaginary part, the output of your function rotates in the complex plane as time ticks forward. When the real part of S is negative, the magnitude is decaying towards zero over time. But if it happened to be positive, that ...