Bernoulli differential equation

In mathematics, an ordinary differential equation of the form

is called a Bernoulli differential equation where is any real number other than 0 or 1.[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 equationEdit

When  , the differential equation is linear. When  , it is separable. In these cases, standard techniques for solving equations of those forms can be applied. For   and  , the substitution   reduces any Bernoulli equation to a linear differential equation. For example, in the case  , making the substitution   in the differential equation   produces the equation  , 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

 

(in this case, more specifically Riccati's equation). The constant function   is a solution. Division by   yields

 

Changing variables gives the equations

 
 
 
 

which can be solved using the integrating factor

 

Multiplying by  ,

 

The 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.[better source needed]

External linksEdit