Homogeneous differential equation.

It's a partial differential equation.

We're solving a differential equation.

What is a differential equation?

We have solved this differential equation.

This was the 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.

What does a homogeneous differential equation mean?

Let's say this is my differential equation.

Multiply it times our original differential equation.

linear differential equation with a constant coefficient.

So let's say this is my differential equation.

Because this is a second order differential equation.

See, they have this as a differential equation.

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

The differential equation must be at least first order

So what is a linear second order differential equation?

Let's solve this differential equation, an interpretation of it.

And we got the solution to the differential equation.

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

And we get a separable second order 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.

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.

Let's apply everything we've learned to an actual differential equation.

Let's say we have the following second order 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.

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

Let's say j is a particular solution to this differential equation.

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.

For example, this differential equation, I don't see an i anywhere here.

And we're asked to find the general solution to this differential equation.

So they claim that this is a solution of this differential equation.

Or as you could call this, a second order ordinary differential equation.

Now, all of the sudden, I have a non linear differential equation.

And this is actually a separable differential equation in and of itself.

It's kind of a sub differential equation to solve our broader one.