# Bernoulli differential equation

In mathematics, an ordinary differential equation of the form:

${\displaystyle y'+P(x)y=Q(x)y^{n}\,}$

is called a Bernoulli differential equation where ${\displaystyle n}$ is any real number and ${\displaystyle n\neq 0}$ and ${\displaystyle n\neq 1}$.[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

Note that for ${\displaystyle n=0}$  and ${\displaystyle n=1}$ , the Bernoulli equation is linear. For ${\displaystyle n\neq 0}$  and ${\displaystyle n\neq 1}$ , the substitution ${\displaystyle u=y^{1-n}}$  reduces any Bernoulli equation to a linear differential equation. For example:

Let's consider the following differential equation: ${\displaystyle x{\frac {dy}{dx}}+y=x^{2}y^{2}}$

Rewriting it in the Bernoulli form (with ${\displaystyle n=2}$ ): ${\displaystyle {\frac {dy}{dx}}+{\frac {1}{x}}y=xy^{2}}$

Now, substituting ${\displaystyle u=y^{-1}}$  we get: ${\displaystyle {\frac {du}{dx}}-{\frac {1}{x}}u=-x}$ , which is a linear differential equation.

## SolutionEdit

Let ${\displaystyle x_{0}\in (a,b)}$  and

${\displaystyle \left\{{\begin{array}{ll}z:(a,b)\rightarrow (0,\infty )\ ,&{\textrm {if}}\ \alpha \in \mathbb {R} \setminus \{1,2\},\\z:(a,b)\rightarrow \mathbb {R} \setminus \{0\}\ ,&{\textrm {if}}\ \alpha =2,\\\end{array}}\right.}$

be a solution of the linear differential equation

${\displaystyle z'(x)=(1-\alpha )P(x)z(x)+(1-\alpha )Q(x).}$

Then we have that ${\displaystyle y(x):=[z(x)]^{\frac {1}{1-\alpha }}}$  is a solution of

${\displaystyle y'(x)=P(x)y(x)+Q(x)y^{\alpha }(x)\ ,\ y(x_{0})=y_{0}:=[z(x_{0})]^{\frac {1}{1-\alpha }}.}$

And for every such differential equation, for all ${\displaystyle \alpha >0}$  we have ${\displaystyle y\equiv 0}$  as solution for ${\displaystyle y_{0}=0}$ .

## ExampleEdit

Consider the Bernoulli equation (more specifically Riccati's equation).[2]

${\displaystyle y'-{\frac {2y}{x}}=-x^{2}y^{2}}$

We first notice that ${\displaystyle y=0}$  is a solution. Division by ${\displaystyle y^{2}}$  yields

${\displaystyle y'y^{-2}-{\frac {2}{x}}y^{-1}=-x^{2}}$

Changing variables gives the equations

${\displaystyle w={\frac {1}{y}}}$
${\displaystyle w'={\frac {-y'}{y^{2}}}.}$
${\displaystyle -w'-{\frac {2}{x}}w=-x^{2}}$
${\displaystyle w'+{\frac {2}{x}}w=x^{2}}$

which can be solved using the integrating factor

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

Multiplying by ${\displaystyle M(x)}$ ,

${\displaystyle w'x^{2}+2xw=x^{4},\,}$

Note that left side is the derivative of ${\displaystyle wx^{2}}$ . Integrating both sides, with respect to ${\displaystyle x}$ , results in the equations

${\displaystyle \int w'x^{2}+2xw\,dx=\int x^{4}\,dx}$
${\displaystyle wx^{2}={\frac {1}{5}}x^{5}+C}$
${\displaystyle {\frac {1}{y}}x^{2}={\frac {1}{5}}x^{5}+C}$

The solution for ${\displaystyle y}$  is

${\displaystyle y={\frac {x^{2}}{{\frac {1}{5}}x^{5}+C}}}$ .

## 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