Exact equations.

Let me write that down.

We're on problem 53.

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.

This is an exact equation.

But anyway, this is exact.

Multiply it times our original differential equation.

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

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

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

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

And let's say we try to do this, and it's not separable, and it's not exact.

This mu of x is when we multiply it, the goal is, after multiplying both sides of the equation by it, we should have an exact equation.

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

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

So it's exact.

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.

And this is the partial, with respect to y.

We had a differential equation that, at least superficially,

And true enough, it was exact.

Make it exact.

And in the next video or the video after that I'll prove the quadratic formula using completing the square.

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

And when we did that, we tested it.

So let me prove the quadratic equation to you, by completing the square.

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

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

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

And I'll leave it up for you to actually substitute back into this original equation.

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

Equation

Equation

All we did is we multiplied both sides of this equation by x.

Welcome back.

These two things are equivalent quantities.

We're asked to use the drop down to form a linear equation with infinitely many solutions.

So it really shouldn't change the solution of that equation, or that differential equation.

And then either factor it or use the actual quadratic equation.

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

Enter an equation

I think you know how to solve them.

So this is not exact.

Polar Equation

Equation Editor

Where's the equation?