Bernoulli differential equation

${\displaystyle y^{\prime }+P(x)y=Q(x)y^n,}$

where ${\displaystyle n}$ is a real number. Some authors allow any real ${\displaystyle n}$, whereas others require that ${\displaystyle n}$ not be 0 or 1. The equation was first discussed in a work of 1695 by Jacob Bernoulli, after whom it is named. The earliest solution, however, was offered by Gottfried Leibniz, who published his result in the same year and whose method is the one still used today.

Bernoulli equations are special because they are nonlinear differential equations with known exact solutions. A notable special case of the Bernoulli equation is the logistic differential equation.

Transformation to a linear differential equation

When ${\displaystyle n=0}$, the differential equation is linear. When ${\displaystyle n=1}$, it is separable. In these cases, standard techniques for solving equations of those forms can be applied. For ${\displaystyle n\ne 0}$ and ${\displaystyle n\ne 1}$, the substitution ${\displaystyle u=y^{1-n}}$ reduces any Bernoulli equation to a linear differential equation

${\displaystyle \frac{du}{dx}-(n-1)P(x)u=-(n-1)Q(x).}$

For example, in the case ${\displaystyle n=2}$, making the substitution ${\displaystyle u=y^{-1}}$ in the differential equation ${\displaystyle \frac{dy}{dx}+\frac{1}{x}y=x{y}^2}$ produces the equation ${\displaystyle \frac{du}{dx}-\frac{1}{x}u=-x}$, which is a linear differential equation.