Differential Equations

So what is a differential equation you might ask? Well, it's an equation that involves derivatives. For example,  dy/dx = 2 is a differential equation. To solve an equation like this, we just use integration, and integrate both sides with respect to x, giving us y = 2x + c. (Don't forget the +c!!!)

Solutions to differential equations involving a + c are called general solutions to the equation. If you're given more information, though, like what y is at a certain point, then you can find a particular solution.

My textbook's also reminding me that, technically, when you integrate both sides, you get a +c on both sides. The integral of (dy/dx) is actually y + c and the integral of 2 is 2x + c. You only need to leave in one of the cs, and this c is technically equal to the difference between the two cs.

Now, if you have something like y(dy/dx) = 2x + 3, you still integrate both sides with respect to x... (and yes, I'm sorry that I still don't know what to type in as a good substitute for the integral sign)

(integral sign) y(dy/dx) dx = (integral sign) 2x + 3 dx
(integral sign) y dy = (integral sign) 2x + 3 dx
(y^2)/2 + c = x^2 + 3x + c
(y^2)/2 = x^2 + 3x + c

Now, look closely at the second line. See how the y is with the dy and the x is with the dx? This is what you should aim to do when you rearrange the equation. Although dy/dx isn't really a fraction- it's the limit of a fraction- thinking of it as one might help you to rearrange the equation so that the ys are with the dy and the xs are with the dx.

Wow, that was a relatively short post (for me, anyway). I can't really be bothered doing area under a curve now because you've probably already done it before and I have other stuff to write about like integrating logarithmic functions, 3D vectors, the shapes of molecules, and Shakespeare's Hamlet. I might write about area under a curve once I've revised everything else that I need to (or maybe while I'm procrastinating over revising those things).

Oh, and by the way...

DON'T FORGET THE + C!!!!!!!!!!!!!!!!!!