But anyway, let's just try to get this in terms of a convolution integral.

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So the convolution theorem well, actually, before I even go to the convolution theorem, let me define what a convolution is.

So if I can figure out the inverse Laplace transform of this, I could at least express our function y of t at least as a convolution integral, even if I don't necessarily solve the integral.

Of this let me pick a nice color of this blue thing in the blue box, and then I can express it as a convolution integral.

What's the convolution of these two things?

Now that we know a little bit about the convolution integral and how it applies to the Laplace transform, let's actually try to solve an actual differential equation using what we know.

Remember, I was doing this in the context of convolution, so

So we just showed you that the convolution if I define

We're taking the convolution of sine of t with cosine of t.

And to kind of give you that comfort, let's actually compute a convolution.

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line integral.

The Gaussian minus exponential distribution is a convolution of a normal distribution with (minus) an exponential distribution.

So the convolution of sine of t with cosine of t is 1 2t sine of t.

And when you do the integration by parts, this could be an indefinite integral, an improper integral, a definite integral, whatever.

In mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions it is important in number theory.

So hopefully, this second example with the convolution to solve an inverse transform clarified things up a little bit.

What is this integral?

This is our integral.

We'd have an integral.

This is an integral.

Press F5, definite integral.

So the convolution of f with g, and this is going to be a function of t, it equals this.

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This integral would be easy.

What would be the integral?

This is an integral sign.

That's this integral right there.

The Transform is an integral.

So anyway, let's take the anti derivative and evaluate this improper definite integral, or this improper integral.

But if you just saw an integral like this, this is the inside of a double integral.

Set here the spatial integral step.

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Anyway, integral from 0 to infinity.

Minus this integral evaluated at 0.

Well, we can take the integral.