Generally, we classify differential equations into two broad categories: partial differential
equations, or PDEs, and ordinary differential equations, or ODEs.
Partial differential equations, or partial derivatives, are what you get when you have
a function defined in terms of multiple variables, but you take the derivative just with respect
to one of them.
For example, you have the equation z=2x^2+3y^4, and you take the partial derivative of z with
respect to x, and the partial derivative of z with respect to y.
You get one partial derivative equation for each variable.
Partial derivatives can be first-order, second-order, or higher-order.
So we just found first-order partial derivatives for z with respect to x and y, because we
took the derivative just once with respect to each variable.
If we wanted to find second-order derivatives, we would take the derivative of the first-order
derivatives.
Because we're dealing with partial derivatives, we can take the partial derivative of both
of these with respect to both variables, to give me four second-order partial derivatives.
As you can see, if we kept going, taking higher-order derivatives, we'd get two first-order partial
derivatives, four second-order partial derivatives, eight third-order partial derivatives, and
so on.
The other broad category of differential equations is ordinary differential equations.
The easiest way to think about ordinary differential equations is to know that they're any equation
that contains something like dy/dx, or d^2y/dx^2.
This does not include d/dx.
d/dx is a verb, or a command that means "take the derivative of whatever follows me."
dy/dx, on the other hand, is a noun that means "the derivative of y with respect to x".
It is literally a derivative function itself.
Ordinary differential equations are generally classified by their "order".
A first-order ordinary differential equation has a first order derivative in it, like dy/dx,
but no higher-order derivatives.
A second-order ordinary differential equation has a second-order derivative in it, like
d^2y/dx^2, but no higher-order derivatives.
In other words, you know the order of the differential equation based on the highest-order
derivative it contains.
Now let's talk about some different types of ordinary differential equations.
Linear, separable, and exact differential equations are types all ordinary differential
equations.
You can have first, second, or higher-order differential equations for these different
types, but let's just focus on the first-order functions.
A first-order linear differential equation looks like this.
You might have an equation that's a linear differential equation and doesn't look exactly
like this, but can be manipulated into this exact form.
Either way, if you can get your equation into this form where you have the derivative here
by itself, a function of x multiplied by y here, and then a function of x over here,
then it's a linear differential equation.
A first-order separable differential equation usually looks like this.
Again, you could easily have one that looks different, but the point is to see whether
or not you can separate the variables.
If you can separate the variables, then you know you have a separable differential equation.
In this equation, I can multiply both sides by dx, then multiply both sides by y.
Now I have my y variables separated on the left from my x variables which are all on
the right.
That's how I know for sure that this is a separable differential equation.
And by the way, regular derivatives, the ones we learn about in Calc 1, those are really
just separable differential equations.
Take this simple single variable function from Calc 1.
When you take the derivative, you get dy/dx=2, then you can multiply both sides by dx, and
you've got separated variables, with y's on the left and x's on the right.
So even this simple derivative, which is really the same as y'=2, is a separable differential
equation.
A first-order exact differential equation looks like this.
As you know by now, your equation might not be in this same format to start with, but
if you can get it into this format, then it's an exact differential equation.
What you're looking for is a function in terms of x and y here, a function in terms of x
and y here, multiplied by the derivative dy/dx, and a 0 value over here.
We haven't even talked about homogeneous or nonhomogeneous differential equations, so
this certainly isn't an exhaustive list of every kind of differential equation.
But it's at least an overview of the most common types of differential equations that
you'd study as part of a calculus class, or at the beginning of a differential equations
class.