# .

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.