is called a Bernoulli differential equation where is any real number other than 0 or 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.
Transformation to a linear differential equation
When , the differential equation is linear. When , it is separable. In these cases, standard techniques for solving equations of those forms can be applied. For and , the substitution reduces any Bernoulli equation to a linear differential equation. For example, in the case , making the substitution in the differential equation produces the equation , which is a linear differential equation.
Bernoulli, Jacob (1695), "Explicationes, Annotationes & Additiones ad ea, quae in Actis sup. de Curva Elastica, Isochrona Paracentrica, & Velaria, hinc inde memorata, & paratim controversa legundur; ubi de Linea mediarum directionum, alliisque novis", Acta Eruditorum. Cited in Hairer, Nørsett & Wanner (1993).
Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems, Berlin, New York: Springer-Verlag, ISBN978-3-540-56670-0.