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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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