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Bernoulli differential equation

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.

Contents

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.

SolutionEdit

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  .

ExampleEdit

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

 .

ReferencesEdit

  • 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. http://mathworld.wolfram.com/BernoulliDifferentialEquation.html
  2. ^ y'-2*y/x=-x^2*y^2, Wolfram Alpha, 01-06-2013

External linksEdit