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 other than 0 or 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

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

Solution

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 ${\displaystyle y_{0}=0}$.

Example

Consider the Bernoulli equation

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

(in this case, more specifically Riccati's equation). The constant function ${\displaystyle y=0}$ 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

${\displaystyle u={\frac {1}{y}}}$

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

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

${\displaystyle u'+{\frac {2}{x}}u=x^{2}}$

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 ${\displaystyle M(x)}$,

${\displaystyle u'x^{2}+2xu=x^{4},\,}$

The left side can be represented as the derivative of ${\displaystyle ux^{2}}$. Applying the chain rule and integrating both sides with respect to ${\displaystyle x}$ results in the equations

${\displaystyle \int (ux^{2})'\,dx=\int x^{4}\,dx}$

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

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

The solution for ${\displaystyle y}$ is

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

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.

External links

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