Math, differential equations.

So the differential equations.

That's true with regular equations or differential equations.

Differential equations are something different.

least the solvable differential equations.

making math matter differential equations in action.

Partial differential equations we'll get into later.

We'll now move from the world of first order differential equations to the world of second order differential equations.

Let's do some more examples with exact differential equations.

Laplace transforms are extremely useful for solving differential equations.

We're talking about books on hypergeometric partial differential equations.

Right now we're doing very basic differential equations, but that's something to keep in mind the whole time you learn differential equations.

And this first class of differential equations I'll introduce you to, they're called separable equations.

Welcome to the video on the introduction to differential equations.

We're going to start embarking on higher order differential equations.

That's what this playlist will deal with, ordinary differential equations.

And even within differential equations, we'll learn later there's a different type of homogeneous differential equation.

We first captured the dynamics with a set of differential equations.

So second order linear homogeneous because they equal 0 differential equations.

To 1,000 decimal places, while solving differential equations in his head.

Now, we can use this knowledge, which is the chain rule using partial derivatives, and this knowledge to now solve a certain class of differential equations, first order differential equations, called exact equations.

Well actually, there's a first big one, ordinary and partial differential equations.

And if you're taking differential equations, it might be on an exam.

Let's do one more homogeneous differential equation, or first order homogeneous differential equation, to differentiate it from the homogeneous linear differential equations we'll do later.

But anyway, for this purpose, I'm going to show you homogeneous differential equations.

And it's just another method for solving a certain type of differential equations.

And in general, one thing that's kind of fun about differential equations, but kind of not as satisfying about differential equations, is it really is just a whole hodgepodge of tools to solve different types of equations.

You could spend 10 years learning physics and you write down the Navier Stokes equations the differential equations of fluid dynamics.

Those are called homogeneous linear differential equations, but they mean something actually quite different.

That makes sense, because the separable differential equations are really just implicit derivatives backwards.

There isn't just one tool or one theory that will solve all differential equations.

This differential equations problem was literally just a problem in using the quadratic equation.

And that applies even when you start doing partial differential equations, et cetera, et cetera.

And I'm getting these problems from page 80 of my old college differential equations books.

I'll actually show you how it is useful to use it to solve differential equations.

We'll actually solve some of the differential equations we did before, using the previous methods.

Now within ordinary differential equations, there's two ways of classifying, and they kind of overlap.

In total, we are talking about 120 variables in a dynamic system of differential equations.

And I haven't made the connection yet on how these second order differential equations are related to the first order ones that I just introduced to these other homogeneous differential equations I introduced you to.

But anyway, these are useful properties to maybe internalize for second order homogeneous linear differential equations.

I think it's reasonable to do one more separable differential equations problem, so let's do it.

I'm just trying to show you as many examples as possible of solving exact differential equations.

Welcome to this first video, and actually the first video in the playlist on differential equations.

A lot of what you'll learn in differential equations is really just different bags of tricks.

The whole point in learning differential equations is that eventually we want to model real physical systems.