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| In [[mathematics]], an [[ordinary differential equation]] of the form
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| : <math>y'+ P(x)y = Q(x)y^n\,</math>
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| is called a '''Bernoulli equation''' when n≠1, 0, which is named after [[Jacob Bernoulli]], who discussed it in 1695 {{harv|Bernoulli|1695}}. Bernoulli equations are special because they are nonlinear differential equations with known exact solutions.
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| ==Solution==
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| Let <math>x_0 \in (a, b)</math> and
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| :<math>\left\{\begin{array}{ll}
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| z: (a,b) \rightarrow (0, \infty)\ ,&\textrm{if}\ \alpha\in \mathbb{R}\setminus\{1,2\},\\
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| z: (a,b) \rightarrow \mathbb{R}\setminus\{0\}\ ,&\textrm{if}\ \alpha = 2,\\\end{array}\right.</math>
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| by a solution of the linear differential equation
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| :<math>z'(x)=(1-\alpha)P(x)z(x) + (1-\alpha)Q(x).</math>
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| Then we have that <math>y(x) := [z(x)]^{\frac{1}{1-\alpha}}</math> is a solution of
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| :<math>y'(x) = P(x)y(x) + Q(x)y^\alpha(x)\ ,\ y(x_0) = y_0 := [z(x_0)]^{\frac{1}{1-\alpha}}.</math>
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| And for every such differential equation, for all <math>\alpha>0</math> we have <math>y\equiv 0</math> as solution for <math>y_0=0</math>.
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| ==Example==
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| Consider the Bernoulli equation (more specifically [[Riccati's equation]]).<ref>[http://www.wolframalpha.com/input/?i=y%27-2*y%2Fx%3D-x^2*y^2 y'-2*y/x=-x^2*y^2], Wolfram Alpha, 01-06-2013</ref>
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| :<math>y' - \frac{2y}{x} = -x^2y^2</math>
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| We first notice that <math>y=0</math> is a solution.
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| Division by <math>y^2</math> yields
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| :<math>y'y^{-2} - \frac{2}{x}y^{-1} = -x^2</math>
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| Changing variables gives the equations
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| :<math>w = \frac{1}{y}</math>
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| :<math>w' = \frac{-y'}{y^2}.</math>
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| :<math>w' + \frac{2}{x}w = x^2</math>
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| which can be solved using the integrating factor
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| :<math>M(x)= e^{2\int \frac{1}{x}dx} = e^{2\ln x} = x^2.</math>
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| Multiplying by <math>M(x)</math>,
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| :<math>w'x^2 + 2xw = x^4,\,</math>
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| Note that left side is the [[derivative]] of <math>wx^2</math>. Integrating both sides results in the equations
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| :<math>\int d[wx^2] = \int x^4 dx</math>
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| :<math>wx^2 = \frac{1}{5}x^5 + C</math>
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| :<math>\frac{1}{y}x^2 = \frac{1}{5}x^5 + C</math>
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| The solution for <math>y</math> is
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| :<math>y = \frac{x^2}{\frac{1}{5}x^5 + C}</math>
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| as well as <math>y=0</math>.
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| == References ==
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| * {{Citation | last1=Bernoulli | first1=Jacob | author1-link=Jacob Bernoulli | title=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 | year=1695 | journal=[[Acta Eruditorum]]}}. Cited in {{harvtxt|Hairer|Nørsett|Wanner|1993}}.
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| * {{Citation | last1=Hairer | first1=Ernst | last2=Nørsett | first2=Syvert Paul | last3=Wanner | first3=Gerhard | title=Solving ordinary differential equations I: Nonstiff problems | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-56670-0 | year=1993}}.
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| <references/>
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| == External links ==
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| * {{planetmath reference|id=7032|title=Bernoulli equation}}
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| * {{planetmath reference|id=2629|title=Differential equation}}
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| * {{planetmath reference|id=7023|title=Index of differential equations}}
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| [[Category:Ordinary differential equations]]
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