Separable Equations

By SkytrailGroup Mathematics Editorial Team April 12, 2026

Method

Rewrite as \(g(y)dy=f(x)dx\), integrate both sides, solve for y.

Examples

Example 1. dy/dx = xy.

Solution. dy/y = x dx → ln|y|=x²/2+C → y=Ae^{{x²/2}}.

In Depth

A separable ODE has the form \(dy/dx = f(x)g(y)\). The solution method: separate variables to get \(dy/g(y) = f(x)\,dx\), then integrate both sides. This works whenever \(g(y)\neq0\); solutions where \(g(y)=0\) are equilibrium (constant) solutions.

Exponential growth and decay \(dy/dt = ky\) is the simplest separable equation. Separating: \(dy/y = k\,dt\), integrating: \(\ln|y| = kt+C\), so \(y = Ae^{kt}\). For \(k>0\): exponential growth (population, compound interest). For \(k<0\): exponential decay (radioactive decay, drug elimination).

The logistic equation \(dy/dt = ry(1-y/K)\) is separable. Partial fractions give the solution \(y(t) = K/(1+Ae^{-rt})\), an S-shaped curve approaching the carrying capacity \(K\). This models population growth with resource limits, epidemic spread, and technology adoption.

Newton's law of cooling \(dT/dt = -k(T-T_\infty)\) is separable: \(T(t) = T_\infty + (T_0-T_\infty)e^{-kt}\). It models the temperature of an object cooling toward ambient temperature \(T_\infty\), with applications in forensics (estimating time of death) and engineering (thermal management).

Key Properties & Applications

A separable ODE \(dy/dx=f(x)g(y)\) is solved by separating variables: \(dy/g(y)=f(x)\,dx\), then integrating both sides. This gives an implicit solution \(G(y)=F(x)+C\) where \(G'=1/g\) and \(F'=f\). Solving for \(y\) (if possible) gives the explicit solution.

Equilibrium solutions occur where \(g(y)=0\) — the function \(y=y_0\) (constant) satisfies the ODE. Stability analysis determines whether nearby solutions approach or diverge from equilibrium: if \(g'(y_0)<0\) (in the autonomous case \(dy/dx=g(y)\)), the equilibrium is stable.

Classic separable equations: exponential growth/decay \(dy/dt=ky\) (solution \(y=Ce^{kt}\)); logistic growth \(dy/dt=ky(1-y/K)\) (solution via partial fractions); Newton's law of cooling \(dT/dt=-k(T-T_\infty)\). These model population dynamics, radioactive decay, and heat transfer.