Exact equations.

Let me write that down.

If it's not separable, but you can still put it in this form, you say, hey, is it an exact equation?

totally fair.

Multiply it times our original differential equation.

I will see you in the next video, where we will actually solve some exact equations See

You could say that's M, or that's N.

We said, well, if we multiply both sides of this equation by mu of x is equal to x, it should make this into an exact differential equation.

So let's put that aside and let's explore another property of partial derivatives, and then we're ready to get the intuition behind exact equations.

One, trying to figure out whether the equations are exact.

This is an exact equation.

But anyway, this is exact.

But if you look here, you actually could have figured out that this is actually a separable equation.

It was different than N with respect to x.

We had a differential equation that, at least superficially,

Anyway, so I'm just going to give a bunch of equations.

I continued to create comics.

So M sub y, or the partial with respect to y, is equal to the partial with respect to x.

And when we did that, we tested it.

Here's a game I would like to play with you. I drew on the right side six different expressions, and I'm going to derive this for you,

One, how do I know that it's an exact equation?

And we're going to use this property in the next series of videos to understand exact equations a little bit more.

Totally, totally.

We have written this in the form that we need, in this form.

We know how to convert fractions to percentages.

But if you see this form, you say, boy, maybe it's an exact equation.

Totally dead. Way dead.

And this is the partial, with respect to y.

And true enough, it was exact.

Make it exact.

Now, why does this happen?

So essentially we want this, we want the left hand side to have this form.

And, in the last video, we had this differential equation.

I would say a lot easier than what we did in the previous first order homogeneous difference equations, or the exact equations.

Last night was a perfectly normal night.

This partial with respect to y, is this, times y prime.

So then this will also be equal.

I mean, I don't remember you having this one. Old blondie over here.

And we knew that it involved this square.

So how do we do that?

So when you complete the square you want the thing to just look like a you want whatever's on the left hand side to look like a perfect square.

So if my tax rate is 30 and I think I can get 10 percent on the stock market, I should actually put a 7 here.

So it's exact.

Let's see.

You would risk its utter ruin?