Bernoulli differential equation
In mathematics, an ordinary differential equation of the form
is called a Bernoulli equation when n≠1, 0, which 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.
Solution
Let and
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
.
Example
Consider the Bernoulli equation (more specifically Riccati's equation).[1]
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
.
References
- 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