Homogeneous differential equation.

Partial differential equations we'll get into later.

We're solving a differential equation.

What is a differential equation?

So what is a differential equation?

We have solved this differential equation.

This was the original differential equation.

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

What does a homogeneous differential equation mean?

Let's say this is my differential equation.

Multiply it times our original differential equation.

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.

So let's say this is my differential equation.

So, for example, this is a differential equation.

So how do we solve this differential equation?

You're saying what function satisfies this differential equation.

See, they have this as a differential equation.

The differential equation must be at least first order

So what is a linear second order differential equation?

But let's solve this properly as a differential equation.

And we got the solution to the differential equation.

Let's solve another 2nd order linear homogeneous differential equation.

An ordinary differential equation is what I wrote down.

We had a differential equation that, at least superficially,

So we could rewrite our differential equation like this.

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

And in order for this to have been an exact differential equation, the partial of this with respect to y would have to equal the partial of this with respect to x.

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

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

Well, a differential equation is an equation that involves an unknown function and its derivatives.

And this is the general solution of this differential equation.

So this is also a solution to the differential equation.

So this is the general solution to this differential equation.

So what is the particular solution to this differential equation?

So the first question is what is a differential equation?

Here, for a differential equation, the solution is a function.

So this was also a solution to this differential equation.

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

Resolving a differential equation means finding the functions that satisfies it.

You actually need two initial conditions to solve this differential equation.

So anyway, this was the general solution to this differential equation.

So there is a certain shape which causes the brightness variation in the image and turns out that he derived partial differential equation which could be solved.

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

Well, let's substitute this whole thing into our original differential equation, right?

And that would be the particular solution, then, for this differential equation.