Bernoulli differential equation

In mathematics, an ordinary differential equation of the form

\( y'+ P(x)y = Q(x)y^n\, \)

is called a Bernoulli equation when n≠1, 0, which is named after Jakob Bernoulli, who discussed it in 1695 (Bernoulli 1695). Bernoulli equations are special because they are nonlinear differential equations with known exact solutions.

Solution

Dividing by \( y^n \) yields

\( \frac{y'}{y^{n}} + \frac{P(x)}{y^{n-1}} = Q(x). \)

A change of variables is made to transform into a linear first-order differential equation.

\( w=\frac{1}{y^{n-1}} \)
\( w'=\frac{(1-n)}{y^{n}}y' \)
\( \frac{w'}{1-n} + P(x)w = Q(x) \)

The substituted equation can be solved using the integrating factor

\( M(x)= e^{(1-n)\int P(x)dx}. \)

Example

Consider the Bernoulli equation

\( y' - \frac{2y}{x} = -x^2y^2 \)

We first notice that y=0 is a solution. Division by \( y^2 \) yields

\( y'y^{-2} - \frac{2}{x}y^{-1} = -x^2 \)

Changing variables gives the equations

\( w = \frac{1}{y} \)
\( w' = \frac{-y'}{y^2}. \)
\( w' + \frac{2}{x}w = x^2 \)

which can be solved using the integrating factor

\( M(x)= e^{2\int \frac{1}{x}dx} = e^{2\ln x} = x^2. \)

Multiplying by M(x),

\( w'x^2 + 2xw = x^4,\, \)

Note that left side is the derivative of \( wx^2 \). Integrating both sides results in the equations

\( \int (wx^2)' dx = \int x^4 dx \)
\( wx^2 = \frac{1}{5}x^5 + C \)
\( \frac{1}{y}x^2 = \frac{1}{5}x^5 + C \)

The solution for y is

\( y = \frac{5x^2}{x^5 + C} \)

as well as y=0.

Verifying using MATLAB symbolic toolbox by running

\( x = dsolve('Dy-2*y/x=-x^2*y^2','x') \)

gives both solutions:

0
\( x^2/(x^5/5 + C1) \)

also see a solution by WolframAlpha, where the trivial solution y=0 is missing.
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.

External links

Bernoulli equation, PlanetMath.org.
Differential equation, PlanetMath.org.
Index of differential equations, PlanetMath.org.

Mathematics Encyclopedia