# Bernoulli equation

From Encyclopedia of Mathematics

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://www.encyclopediaofmath.org/index.php?title=Bernoulli_equation&oldid=40764

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