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 and $n\neq 0$ or ${\displaystyle n\neq 1}$.^[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 equation

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 $n=2$): ${\displaystyle {\frac {dy}{dx}}+{\frac {1}{x}}y=xy^{2}}$

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

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 (more specifically Riccati's equation).^[2]

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

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

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

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

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

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

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 $M(x)$,

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

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

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

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

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

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.

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

External links

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