Bernoulli differential equation

In mathematics, an ordinary differential equation of the form:

is called a Bernoulli differential equation where ${\displaystyle n}$ is any real number and ${\displaystyle n\neq 0}$ and ${\displaystyle n\neq 1}$. It is named after Jacob Bernoulli who discussed it in 1695. Bernoulli equations are special because they are nonlinear differential equations with known exact solutions. A famous special case of the Bernoulli equation is the logistic differential equation.

Note that for ${\displaystyle n=0}$ and ${\displaystyle n=1}$, the Bernoulli equation is linear. For ${\displaystyle n\neq 0}$ and ${\displaystyle n\neq 1}$, the substitution ${\displaystyle u=y^{1-n}}$ reduces any Bernoulli equation to a linear differential equation. For example:

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