Khanh at 04:09, 6 November 2021

2021-11-06T04:09:42Z

Lila at 21:54, 14 October 2021

2021-10-14T21:54:56Z

Lila at 21:54, 14 October 2021

2021-10-14T21:54:27Z

Johnraymond.yanez: Created page with "[https://tutorial.math.lamar.edu/Classes/DE/Bernoulli.aspx Bernoulli Equations Notes]. Produced by Paul Dawkins, Lamar University"

2020-08-12T13:30:26Z

Created page with "[https://tutorial.math.lamar.edu/Classes/DE/Bernoulli.aspx Bernoulli Equations Notes]. Produced by Paul Dawkins, Lamar University"

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[https://tutorial.math.lamar.edu/Classes/DE/Bernoulli.aspx Bernoulli Equations Notes]. Produced by Paul Dawkins, Lamar University

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 ==Resources==
 * [https://tutorial.math.lamar.edu/Classes/DE/Bernoulli.aspx Bernoulli Equations Notes]. Produced by Paul Dawkins, Lamar University
-* [https://en.wikipedia.org/wiki/Bernoulli_differential_equation Bernoulli differential equation], Wikipedia

← Older revision		Revision as of 21:54, 14 October 2021
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	==Resources==		==Resources==
	* [https://tutorial.math.lamar.edu/Classes/DE/Bernoulli.aspx Bernoulli Equations Notes]. Produced by Paul Dawkins, Lamar University		* [https://tutorial.math.lamar.edu/Classes/DE/Bernoulli.aspx Bernoulli Equations Notes]. Produced by Paul Dawkins, Lamar University
		+	* [https://en.wikipedia.org/wiki/Bernoulli_differential_equation Bernoulli differential equation], Wikipedia

← Older revision		Revision as of 21:54, 14 October 2021
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−	[https://tutorial.math.lamar.edu/Classes/DE/Bernoulli.aspx Bernoulli Equations Notes]. Produced by Paul Dawkins, Lamar University	+	In mathematics, an ordinary differential equation is called a '''Bernoulli differential equation''' if it is of the form
		+
		+	: <math>y'+ P(x)y = Q(x)y^n,</math>
		+
		+	where <math>n</math> is a real number. Some authors allow any real <math>n</math>, whereas others require that <math>n</math> not be 0 or 1. The equation was first discussed in a work of 1695 by Jacob Bernoulli, after whom it is named. The earliest solution, however, was offered by Gottfried Leibniz, who published his result in the same year and whose method is the one still used today.
		+
		+	Bernoulli equations are special because they are nonlinear differential equations with known exact solutions. A notable special case of the Bernoulli equation is the logistic differential equation.
		+
		+	==Transformation to a linear differential equation==
		+	When <math> n = 0</math>, the differential equation is linear. When <math>n = 1</math>, it is separable. In these cases, standard techniques for solving equations of those forms can be applied. For <math>n \neq 0</math> and <math>n \neq 1</math>, the substitution <math>u = y^{1-n} </math> reduces any Bernoulli equation to a linear differential equation
		+	: <math>\frac{du}{dx} - (n-1)P(x)u = - (n-1)Q(x).</math>
		+	For example, in the case <math>n = 2</math>, making the substitution <math>u=y^{-1}</math> in the differential equation <math> \frac{dy}{dx} + \frac{1}{x}y=xy^2 </math> produces the equation <math>\frac{du}{dx} -\frac{1}{x}u=-x</math>, which is a linear differential equation.
		+
		+	==Solution==
		+	Let <math>x_0 \in (a, b)</math> and
		+	:<math>\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.</math>
		+	be a solution of the linear differential equation
		+	:<math>z'(x)=(1-\alpha)P(x)z(x) + (1-\alpha)Q(x).</math>
		+	Then we have that <math>y(x) := [z(x)]^{\frac{1}{1-\alpha}}</math> is a solution of
		+	:<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>
		+	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>.
		+
		+	==Example==
		+	Consider the Bernoulli equation
		+	:<math>y' - \frac{2y}{x} = -x^2y^2</math>
		+	(in this case, more specifically Riccati's equation).
		+	The constant function <math>y=0</math> is a solution.
		+	Division by <math>y^2</math> yields
		+	:<math>y'y^{-2} - \frac{2}{x}y^{-1} = -x^2</math>
		+	Changing variables gives the equations
		+	:<math>\begin{align}
		+	u = \frac{1}{y} \; & , ~ u' = \frac{-y'}{y^2} \\
		+	-u' - \frac{2}{x}u &= - x^2 \\
		+	u' + \frac{2}{x}u &= x^2
		+	\end{align}</math>
		+	which can be solved using the integrating factor
		+	:<math>M(x)= e^{2\int \frac{1}{x}\,dx} = e^{2\ln x} = x^2.</math>
		+	Multiplying by <math>M(x)</math>,
		+	:<math>u'x^2 + 2xu = x^4.</math>
		+
		+	The left side can be represented as the derivative of <math>ux^2</math> by reversing the product rule. Applying the chain rule and integrating both sides with respect to <math>x</math> results in the equations
		+	:<math>\begin{align}
		+	\int \left(ux^2\right)' dx &= \int x^4\,dx \\
		+	ux^2 &= \frac{1}{5}x^5 + C \\
		+	\frac{1}{y}x^2 &= \frac{1}{5}x^5 + C
		+	\end{align}</math>
		+	The solution for <math>y</math> is
		+	:<math>y = \frac{x^2}{\frac{1}{5}x^5 + C}.</math>
		+
		+
		+	==Resources==
		+	* [https://tutorial.math.lamar.edu/Classes/DE/Bernoulli.aspx Bernoulli Equations Notes]. Produced by Paul Dawkins, Lamar University

Bernoulli Equations (1st Order) - Revision history

Khanh at 04:09, 6 November 2021

Lila at 21:54, 14 October 2021

Lila at 21:54, 14 October 2021

Johnraymond.yanez: Created page with "[https://tutorial.math.lamar.edu/Classes/DE/Bernoulli.aspx Bernoulli Equations Notes]. Produced by Paul Dawkins, Lamar University"