Bernoulli differential equation

In mathematics, an ordinary differential equation of the form

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

is called a Bernoulli differential equation where $n$ is any real number other than 0 or 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 equation

When $n=0$, the differential equation is linear. When $n=1$, it is separable. In these cases, standard techniques for solving equations of those forms can be applied. For $n\neq 0$ and $n\neq 1$, the substitution $u=y^{1-n}$ reduces any Bernoulli equation to a linear differential equation. For example, in the case $n=2$, making the substitution $u=y^{-1}$ in the differential equation ${\frac {dy}{dx}}+{\frac {1}{x}}y=xy^{2}$ produces the equation ${\frac {du}{dx}}-{\frac {1}{x}}u=-x$, which is a linear differential equation.

Solution

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

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

be a solution of the linear differential equation

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

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

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

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

Example

Consider the Bernoulli equation

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

(in this case, more specifically Riccati's equation). The constant function $y=0$ is a solution. Division by $y^{2}$ yields

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

Changing variables gives the equations

$w={\frac {1}{y}}$

$w'={\frac {-y'}{y^{2}}}.$

$-w'-{\frac {2}{x}}w=-x^{2}$

$w'+{\frac {2}{x}}w=x^{2}$

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 $M(x)$,

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

The left side is the derivative of $wx^{2}$. Integrating both sides with respect to $x$ results in the equations

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

$wx^{2}={\frac {1}{5}}x^{5}+C$

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

The solution for $y$ is

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

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.

External links

• "Bernoulli equation". PlanetMath.
• "Differential equation". PlanetMath.
• "Index of differential equations". PlanetMath.