View source for Bernoulli equation

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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 [[#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 $y$ is considered as the independent variable, while $x$ is an unknown function of $y$.

====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>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.L. Ince,   "Ordinary differential equations" , Dover, reprint  (1956)</TD></TR></table>