One thing that's always bothered me is the idea of a taking the derivative of a variable when that variable isn't a function. For example, the differential of an expression x^2 is 2x dx. But when x is given an a concrete value, let's take x=10, we say that the differential of 10 is 0 (because it's a constant). So, 2x dx = 2(10) d(10) = 20 * 0 = 0. In fact, for any value we pick for x, dx will vanish to 0. Now the reason we can get the right answer is because somewhere under the hood, we're talking about limits and because of that we can get away with dividing 0 by 0. But there's too much hand waving and not enough rigor for my mathematical blood! What I want to know is... is there's a prettier way to formulate differential equations? What would you do to put them on a solid mathematical basis? I've been wondering about this for a week or two, and the best idea I had was coming up with an idea that you could "promote" those boring variables to parametrized functions. Then, instead of relating their values (x = y^2+1), you'd relate their values when applied an argument (x(t) = y(t)^2+1). Then, the differential operation would be something analogous to the derivative with respect to the parametrized variable, t.... or something like that.