Bernoulli differential equation

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In mathematics, an ordinary differential equation of the form:

is called a Bernoulli differential equation where is any real number and and .[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[edit]

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.


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 .


Consider the Bernoulli equation (more specifically Riccati's equation).[2]

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, with respect to , 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 .
  1. ^ Weisstein, Eric W. "Bernoulli Differential Equation." From MathWorld--A Wolfram Web Resource.
  2. ^ y'-2*y/x=-x^2*y^2, Wolfram Alpha, 01-06-2013

External links[edit]