A first-order ODE is separable when it can be written so that the dependent variable and independent variable appear on opposite sides:

$$\frac{dy}{dx}=g(x)h(y).$$

Where division by $h(y)$ is valid,

$$\frac{1}{h(y)},dy=g(x),dx.$$

Integrating both sides gives an implicit relation

$$\int \frac{1}{h(y)},dy=\int g(x),dx+C.$$

The resulting equation may then be solved for $y$ when convenient.

Solutions lost by dividing by $h(y)$ must be checked separately. In particular, values for which $h(y)=0$ may give constant equilibrium solutions.

Separation of variables is a structural method: the key step is recognizing when all $y$-dependence and all $x$-dependence can be isolated before integrating.