Bernoulli differential equation
From Wikipedia, the free encyclopedia
In mathematics, an ordinary differential equation of the form
is called a Bernoulli equation when n≠1, 0. Bernoulli equations are special because they are nonlinear differential equations with known exact solutions. Dividing by yn yields
A change of variables is made to transform into a linear first-order differential equation.
The substituted equation can be solved using the integrating factor
The Bernoulli equation is named after Jakob Bernoulli, who discussed it in 1695 (Bernoulli 1695).
[edit] Example
Consider the Bernoulli equation
Division by y2 yields
Changing variables gives the equations
which can be solved using the integrating factor
Multiplying by M(x),
Note that left side is the derivative of wx2. Integrating both sides results in the equations
The solution for y is
[edit] References
- Bernoulli, Jacob (1695), "Explicationes, Annotationes & Additiones ad ea, quae in Actis sup. anni 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.
