is called a Bernoulli differential equation where is any real number and and . 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 equationEdit
Note that for and , the Bernoulli equation is linear. For and , the substitution reduces any Bernoulli equation to a linear differential equation. For example:
Let's consider the following differential equation:
Rewriting it in the Bernoulli form (with ):
Now, substituting we get: , 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.