Bernoulli differential equation

In mathematics, an ordinary differential equation is called a Bernoulli differential equation if it is of the form

where is a real number. Some authors allow any real ,[1][2] whereas others require that not be 0 or 1.[3][4] The equation was first discussed in a work of 1695 by Jacob Bernoulli, after whom it is named. The earliest solution, however, was offered by Gottfried Leibniz, who published his result in the same year and whose method is the one still used today.[5]

Bernoulli equations are special because they are nonlinear differential equations with known exact solutions. A notable special case of the Bernoulli equation is the logistic differential equation.

Transformation to a linear differential equation

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.

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

(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 can be represented as the derivative of . Applying the chain rule and integrating both sides with respect to results in the equations

The solution for is

.

Notes

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.
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