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A separable differential equation is an ordinary differential equation that can be separated into two integrals; that is, in the form

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

They are arguably the simplest ODEs to solve, as they will always have the solution

$\int\dfrac{dy}{g(y)}=\int f(x)dx$

If $f(x)$ is equal to some constant, the DE is an autonomous differential equation. For example:

$\frac{dy}{dx}=x^2y$
$\frac{dy}{y}=x^2dx$
$\int\dfrac{dy}{y}=\int x^2dx$
$\ln(|y|)+C_1=\frac{x^3}{3}+C_2$
$\ln(|y|)=\frac{x^3}{3}+C_3$
$|y|=e^{\frac{x^3}{3}+C_3}=C_4e^{\frac{x^3}{3}}$
$y=\pm C_4e^{\frac{x^3}{3}}$