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Difference between revisions of "Bernoulli equation"

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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Bernoulli,  ''Acta Erud.''  (1695)  pp. 59–67; 537–557</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Kamke,  "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''1. Gewöhnliche Differentialgleichungen''' , Chelsea, reprint  (1971)</TD></TR></table>
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<table>
 
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<TR><TD valign="top">[1]</TD> <TD valign="top">  J. Bernoulli,  ''Acta Erud.''  (1695)  pp. 59–67; 537–557</TD></TR>
 
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<TR><TD valign="top">[2]</TD> <TD valign="top">  E. Kamke,  "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''1. Gewöhnliche Differentialgleichungen''' , Chelsea, reprint  (1971)</TD></TR>
 
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  E.L. Ince,  "Ordinary differential equations" , Dover, reprint  (1956)</TD></TR>
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====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.L. Ince,  "Ordinary differential equations" , Dover, reprint  (1956)</TD></TR></table>
 

Latest revision as of 18:48, 9 January 2024

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)
[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=54950
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article