Separable Differential Equations

A rigorous method for separable equations, including equilibrium solutions, implicit forms, and maximal intervals.

Classify → choose method → apply condition. A differential equation is classified by order, linearity, and separability before a method is selected. The final condition determines constants and a residual check tests the solution.

Precise definition

An equation is separable if it can be written $y'=g(x)h(y)$. For non-equilibrium portions where $h(y)\ne0$, rearrange $dy/h(y)=g(x)dx$ and integrate. Any root $h(y_)=0$ gives a constant solution $y=y_$ that division would remove.

Notation and mathematical language

The differential notation abbreviates a justified chain-rule computation. If $H'(y)=1/h(y)$ and $G'(x)=g(x)$, then $d[H(y(x))]/dx=g(x)$, so $H(y)=G(x)+C$. The result may remain implicit.

Conceptual picture

Separation accumulates reciprocal state-dependent rate on one side and independent-variable effect on the other. Equilibria form barriers under uniqueness. An implicit formula can encode multiple branches, so initial data identify the relevant one.

Fully worked example

Interpretation and application

Separable models include growth, decay, drag, and reaction kinetics. Exact separation solves the stated equation; it does not establish that the product-form rate law is causal or valid outside observed conditions.