Difference between revisions of "Bernoulli equation"
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An ordinary first-order differential equation | An ordinary first-order differential equation | ||
| − | + | $$a_0(x)y'+a_1(x)y=f(x)y^\alpha,$$ | |
| − | where | + | 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 |
| − | + | $$[f(y)x+g(y)x^\alpha]y'=h(y),\quad\alpha\neq0,1,$$ | |
| − | are also Bernoulli equations if | + | are also Bernoulli equations if $y$ is considered as the independent variable, while $x$ is an unknown function of $y$. |
====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> |
| − | + | <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> | |
| − | + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> E.L. Ince, "Ordinary differential equations" , Dover, reprint (1956)</TD></TR> | |
| − | + | </table> | |
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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=15844
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