Method of separation of variables

In brief — Solution method

Using Leibniz’s notation $y' = \dfrac{dy}{dx}$, we “separate”: $\frac{dy}{dx} = f(x)\cdot g(y) \implies \frac{dy}{g(y)} = f(x)\,dx.$ Integrating both sides: $\int\frac{dy}{g(y)} = \int f(x)\,dx.$ The constant of integration $C$ is determined by an initial condition $y(x_0)=y_0$ (Cauchy problem).

Remark — Leibniz's "trick" for ODEs

Separating $\dfrac{dy}{dx} = f(x)g(y)$ into $\dfrac{dy}{g(y)} = f(x)\,dx$ is legitimate precisely because Leibniz’s notation treats $dy$ and $dx$ as “fractions” — the same principle seen for the chain rule and for substitution in integrals. It all fits together.

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Topics: Differential equations
Concepts: Initial condition · Cauchy problem · Separable variables
Skills: Integrating · Solving equations
People: Augustin-Louis Cauchy · Gottfried Leibniz