Bernoulli differential equation
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is called a Bernoulli equation when n≠1, 0. It is named after Jacob Bernoulli who discussed it in 1695 (Bernoulli 1695). Bernoulli equations are special because they are nonlinear differential equations with known exact solutions. A famous special case of the Bernoulli differential equation is the logistic differential equation.
be a solution of the linear differential equation
Then we have that is a solution of
And for every such differential equation, for all we have as solution for .
We first notice that is a solution. Division by yields
Changing variables gives the equations
which can be solved using the integrating factor
Multiplying by ,
Note that left side is the derivative of . Integrating both sides results in the equations
The solution for is
- 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, ISBN 978-3-540-56670-0.
- y'-2*y/x=-x^2*y^2, Wolfram Alpha, 01-06-2013