A first-order equation is separable when you can write $\frac{dy}{dx}=g(x)h(y)$ (or equivalently $M(x)N(y)\,dx+P(x)Q(y)\,dy=0$ in differential form). The method is: move $y$-factors to one side and $x$-factors to the other, then integrate.

The steps

  1. Write $\frac{dy}{dx}=g(x)h(y)$.
  2. If $h(y)\neq 0$, rewrite $\frac{1}{h(y)}\,dy = g(x)\,dx$.
  3. Integrate both sides: $\int \frac{1}{h(y)}\,dy = \int g(x)\,dx + C$ (absorb constants sensibly).
  4. Solve for $y$ if you can (explicit solution); otherwise leave an implicit relation.
  5. Check singular solutions: any roots of $h(y)=0$ that make $y$ constant may be solutions lost when you divided.
  6. Apply initial conditions to fix $C$ when you have an IVP.

Example Solve $\frac{dy}{dx}=ky$ with $y(0)=y_0$. Separate: $\frac{dy}{y}=k\,dx$ (for $y\neq 0$). Integrate: $\ln|y|=kx+C$, so $y=Ae^{kx}$. The missing solution $y\equiv 0$ corresponds to $A=0$. The IVP yields $y=y_0 e^{kx}$.

Pitfall. Absolute values and domains matter. Extending a local solution through a singularity of $h(y)$ or $g(x)$ can be invalid. Also, not every ODE is separable — forcing the algebra invents nonsense.

Zoom out with Differential equations first look. Refresh derivative meaning in What a derivative actually measures.

Citations & further reading