What the heck is it!?
Relax, it's not as confusing as all that. A differential equation looks a lot like the equations you encounter in plain old algebra -- 2y-4x = 5 and the like. The word "differential" means that one of the variables (x's and y's to you non-mathers) in your equation is a derivative of a function.
If you haven't studied The Calculus -- and who can blame you, really? -- a derivative is a special function derived from another function. If you graph a function on a grid (or in space), and for some reason you want to know what the tangent line (or plane) is at every point of that function, you would compute that function's derivative. This tangent tells you how quickly your function is changing, and in what direction.
Sounds boring, I know, but it's a lot more useful than it sounds. For instance, if you have a function (y) telling you what the position of an object is, the derivative of that function (y' -- say "why-prime") would tell you its speed (how fast it's changing position), and the derivative of that derivative (y'' -- say "why-double-prime") would tell you its acceleration (how fast it's changing speed).
Now, let's go back to our algebra equation:
2y-4x = 5. If you wanted to know what y was, in terms of x, you'd shuffle things around according to what your math teacher told you and come up with y = 2x+5/2. But if we replaced y with its own derivative, and handed you 2y'-4x = 5, you'd be stuck, because you don't know calculus. But if you did, you'd solve for y' -- y' = 2x+5/2 -- and integrate it. That's like finding the derivative, but backwards. The solution happens to be y = x2+5x/2+c, where c is any real number you like. Yes, any real number -- this means that, technically, there are an infinite number of y's for any one value of x. Professional mathematicians have been trying for centuries to get rid of this inconvenience and make their problem-solving much easier, but it looks like we're stuck with it. If you're lucky, you're in a situation where you can set c = 0 and lock things down, but don't bet on it.
That wasn't a differential equation, though. That was just calculus. Differential equations are when you have both y and y' in the same equation -- 2y'-4xy+x = 5, for instance. You can't solve for y with a simple integral this time, but you can solve for it. You'll still have that nasty, inconvenient c in your solution, but if you have what's called an initial condition -- where y equals some number when x is zero -- you can get rid of it.
If your equation has just y and y' in it, it's called a first order differential equation. If you have y'' in it, it's a second order equation. A linear differential equation -- which means y and all its derivatives are multiplied by x's and numbers instead of each other -- is easier to solve than a nonlinear one.
How on earth do I solve one?
I'm not going into detail on this, partly because it would require an entire textbook but mostly because I nearly failed the class myself. The reason I nearly failed the class is because there is no way to solve a differential equation.
Well, that's not exactly true. There's no one, universal method for solving all differential equations. Worse yet, the answer you get may look entirely different from the answer someone else gets, and both of you will still be correct. This makes it very hard to check your work in the back of the textbook, among other things.
However, if your equation happens to be one of a number of special types, then there are tried and true ways to solve them. The hard part is recognizing an equation as a certain type and remembering which types are solved using what techniques.
It's not fun, especially if you hated calculus. Here's a quick step-by-step on how to solve a first order linear differential equation like y'+tan(x)y = cos2(x), which I chose because it's already been solved right here next to me. A first order linear differential equation can always be written in the form y'+f(x)y = g(x), where f(x) and g(x) are some two functions of x, and only x.
First, find something called the integrating factor, which is a fancy term meaning "special number which makes it easier to solve this problem." The integrating factor for these types of equations is e to the power of the integral of whatever you're multiplying by y (in this case, tan(x)). After much flipping through your calculus notes, you're able to simplify this integrating factor to sec(x).
Second, multiply the integrating factor by whatever isn't multiplied by y or y' -- in this case, cos2(x) -- and integrate that. Again, your calculus notes eventually tell you that the integral of sec(x)cos2(x) is sin(x)+c (there's that nasty c again).
Third, divide what you got in the second step by the integrating factor to get y. In our example, we now have y = (sin(x)+c)/sec(x) = sin(x)cos(x)+c*cos(x). If you have an initial condition like, oh, y(0) = 2, you can solve for c: sin(0)cos(0)+c*cos(0) = 2 and, after a little more time with your notes, c=2. Our final answer is y = sin(x)cos(x)+2cos(x), and you can breathe again.
If a differential equation isn't one that can be solved neatly, there are still ways to break it down to a not-so-neat but useable solution. LaPlace transforms and Fourier transforms are the usual ways to go about this.
Why would I want to do this?
You'd be surprised. You probably won't want it in everyday life, of course, but scientists of all sorts run into equations like this in their daily work. Radioactive matter (like plutonium or carbon-14) decays at a rate which changes according to how much hasn't yet decayed, and this can be represented by a linear differential equation. Similarly, the rate of change of an area's population will change depending on how many people are there. Newton's law of cooling is a linear differential equation, too, and can be used to determine how long a body has been dead in a room (along with other, less gory applications).
And then there are the more advanced, not-for-the-faint-of-heart applications that engineers and physicists encounter, like multiresolution analysis, fuzzy logic, electromagnetic fields, cellular growth in an organism, and the engineering of structures that are subject to variable stress like bridges and skyscrapers.
In short, whenever you need to model something that changes depending on how much it's changed, you'll need a differential equation. This actually includes most of the stuff in the universe. With a few special exceptions.
I don't need any of that. I just want to balance my checkbook every month.
Fine with us. Interested in knowing what it took to design the pocket calculator you're using, though?... Hey, come back here!...