separation of variables

Separation of variables is the actual hands-on procedure you run to solve a separable equation. If a separable equation is the type of problem, separation of variables is the recipe you follow — the sequence of algebraic and integration steps that turns dy/dx = f(x) g(y) into an answer.

The steps are simple to state. First, treat dy/dx as a genuine ratio of differentials and rearrange to dy/g(y) = f(x) dx so each variable is alone on its side. Second, place an integral sign in front of both sides and integrate: integral of 1/g(y) dy equals integral of f(x) dx, plus a single arbitrary constant C (one constant covers both sides). Third, if you can, solve the resulting relation for y to get an explicit solution; if you cannot, leave it as an implicit general solution. Throughout, you must separately list any constant solutions where g(y) = 0, because dividing by g(y) hid them.

This is usually the first solution technique anyone learns, and it is worth getting the mechanics clean because the same move — split, integrate, reattach a constant — reappears later (for instance in solving partial differential equations by separating space and time). One honest point: treating dy/dx as a fraction you can split is a convenient shorthand, justified rigorously by the chain rule and substitution; it is a reliable habit here, not a license to manipulate dy and dx however you please in every context.