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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}} $
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