Why Laplace transforms are so useful

I want to show you this simple simulation that I put together that has a mass on a spring, but it's being influenced by an external force that oscillates back and forth. Now, if there was no external force and you pull out this mass and you just let it go, the spring has some kind of natural frequency that it wants to oscillate at. But here, when I'm adding that external force, like a wind blowing back and forth, it oscillates at a distinct unrelated frequency. What I want you to notice is how in the beginning you get this very irregularlook behavior gets kind of stronger and then weaker and then stronger again before eventually it settles into a rhythm.

What specifically is going on there? How could you mathematically analyze what exactly this weird wibbly startup trajectory is? And could you predict how long it takes before the system hits its stride? And when it does hit that stride, could you predict exactly how big the swings back and forth are?

One of the most powerful tools for studying systems like this and many many others is the lelass [music] transform. And today I want to show you exactly how it looks to use this tool to study differential equations and analyze dynamic systems. For context, this is the third chapter in a sequence all about this lelass transform. And as a very quick recap, in the first chapter, you and I became acquainted with functions that look like e to the s * t, where s is a complex number.

And the output of these functions kind of spirals through the complex plane. The key concept you have to have in your mind is what engineers call the s plane [music] which is the complex plane representing all possible values of this term s in the exponent. The idea is to think of each point of the plane as encoding the entire function e to the s * t. And the primary takeaway is that bigger imaginary values of s correspond to functions with more oscillation than negative real parts reflect decay and positive real parts reflect growth.

The reason we care is that many functions in nature can be broken down into exponential pieces. So what we want is a machine that exposes how that breakdown looks. This is the key motivation for what lelass ...