Finding General Solutions Using Separation of Variables

A differential equation gives you a rate, dy/dx, written in terms of x and y mixed together. If you can algebraically untangle that mix so every y (and dy) sits on one side and every x (and dx) sits on the other, you've turned one hard equation into two easy ones. That's the whole trick: dy/dx = f(x)g(y) becomes dy/g(y) = f(x) dx, and now each side is a plain integral with one variable, something you already know how to do.

Once separated, you integrate both sides independently. The left side gives you some function of y, the right side some function of x, plus a constant of integration C. Because that C can be any real number, what you get isn't one curve but an infinite family of curves, all with the same shape but shifted up or down. This is the general solution. If you're handed an initial condition like y(0) = 3, you plug it in right after integrating to solve for the one value of C that picks out the particular curve passing through that point.

This topic is the engine behind everything else in the unit. Slope fields are the picture of the whole family of solutions before you solve anything; separation of variables is how you actually produce the formula behind that picture. And the two big growth models coming next, exponential growth dy/dx = ky and logistic growth, are both just separable equations in disguise, so the algebra you practice here is exactly what unlocks them.