Bernoulli differential equation

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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 Jacob Bernoulli, who discussed it in 1695 (Bernoulli 1695). Bernoulli equations are special because they are nonlinear differential equations with known exact solutions.

Solution

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

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

by 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).^[1]

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

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

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

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

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

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

$\int d[wx^{2}]=\int x^{4}dx$

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

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

The solution for $y$ is

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

as well as $y=0$ .

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 .

↑ y'-2*y/x=-x^2*y^2, Wolfram Alpha, 01-06-2013

External links

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