Separable Equations: Technique and Initial Conditions

A separable equation is a first-order ODE that can be written as $dy/dx = g(x)/h(y)$, putting all $y$-dependence on one side and all $x$-dependence on the other. The solution technique is: move the $y$-terms to the left, $x$-terms to the right, then integrate both sides. This is the single most-used analytical method for first-order ODEs. In finance, separable equations produce the continuous compounding formula $P(t) = P_0 e^{rt}$ (from $dP/dt = rP$), the Black-Scholes terminal distribution (from the log-normal SDE reduced to a separable ODE), and the logistic growth models used in credit risk and commodity capacity models.

Learning objectives. By the end of this note you should be able to: