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Difference between revisions of "Bernoulli equation"

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An ordinary first-order differential equation
 
An ordinary first-order differential equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015600/b0156001.png" /></td> </tr></table>
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$$a_0(x)y'+a_1(x)y=f(x)y^\alpha,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015600/b0156002.png" /> is a real number other than zero or one. This equation was first studied by J. Bernoulli [[#References|[1]]]. The substitution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015600/b0156003.png" /> converts the Bernoulli equation to a linear inhomogeneous first-order equation, [[#References|[2]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015600/b0156004.png" />, the solution of the Bernoulli equation is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015600/b0156005.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015600/b0156006.png" />, at some points the solution is no longer single-valued. Equations of the type
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where $\alpha$ is a real number other than zero or one. This equation was first studied by J. Bernoulli [[#References|[1]]]. The substitution $y^{1-\alpha}=z$ converts the Bernoulli equation to a linear inhomogeneous first-order equation, [[#References|[2]]]. If $\alpha>0$, the solution of the Bernoulli equation is $y\equiv0$; if $0<\alpha<1$, at some points the solution is no longer single-valued. Equations of the type
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015600/b0156007.png" /></td> </tr></table>
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$$[f(y)x+g(y)x^\alpha]y'=h(y),\quad\alpha\neq0,1,$$
  
are also Bernoulli equations if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015600/b0156008.png" /> is considered as the independent variable, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015600/b0156009.png" /> is an unknown function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015600/b01560010.png" />.
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are also Bernoulli equations if $y$ is considered as the independent variable, while $x$ is an unknown function of $y$.
  
 
====References====
 
====References====

Latest revision as of 19:32, 31 March 2017

An ordinary first-order differential equation

$$a_0(x)y'+a_1(x)y=f(x)y^\alpha,$$

where $\alpha$ is a real number other than zero or one. This equation was first studied by J. Bernoulli [1]. The substitution $y^{1-\alpha}=z$ converts the Bernoulli equation to a linear inhomogeneous first-order equation, [2]. If $\alpha>0$, the solution of the Bernoulli equation is $y\equiv0$; if $0<\alpha<1$, at some points the solution is no longer single-valued. Equations of the type

$$[f(y)x+g(y)x^\alpha]y'=h(y),\quad\alpha\neq0,1,$$

are also Bernoulli equations if $y$ is considered as the independent variable, while $x$ is an unknown function of $y$.

References

[1] J. Bernoulli, Acta Erud. (1695) pp. 59–67; 537–557
[2] E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 1. Gewöhnliche Differentialgleichungen , Chelsea, reprint (1971)


Comments

References

[a1] E.L. Ince, "Ordinary differential equations" , Dover, reprint (1956)
How to Cite This Entry:
Bernoulli equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bernoulli_equation&oldid=15844
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article