separable equation

Imagine an equation where everything that mentions x lives in one bag and everything that mentions y lives in another, with nothing tangled together. A separable equation is exactly that lucky case: a first-order equation you can rearrange so all the y-stuff (including the dy) sits on one side and all the x-stuff (including the dx) sits on the other. Once the variables are pulled apart, you can integrate each side on its own and you are essentially done.

Concretely, a separable equation can be written as dy/dx = f(x) g(y) — the right side is a product of a function of x alone and a function of y alone. You move it to the form dy/g(y) = f(x) dx (dividing by g(y)), then integrate both sides. For example, dy/dx = x y separates to dy/y = x dx, and integrating gives ln|y| = x^2/2 + C, so y = A e^(x^2/2). The whole method works because the chain rule, run backwards, lets each side be integrated in its own variable.

Separable equations are the very first solvable family you meet, and a surprising number of real models land here: exponential growth and decay, radioactive decay, Newton's law of cooling, and the logistic equation are all separable. The honest catch is that most differential equations are NOT separable — the variables refuse to come apart — which is why you need linear methods, substitutions, and ultimately numerical tools. There is also a quiet hazard: dividing by g(y) can throw away any solution where g(y) = 0, so watch for lost constant solutions.