# Bernoulli equation

The Bernoulli equation has the form

$\frac{dy}{dx}}+f(x)y=g(x){y}^{k$ | (1) |

where $f$ and $g$ are continuous^{} real functions and $k$ is a ($\ne 0$, $\ne 1$). Such a nonlinear equation (http://planetmath.org/DifferentialEquation) is got e.g. in examining the motion of a by ${y}^{k}$. It yields

${y}^{-k}{\displaystyle \frac{dy}{dx}}+f(x){y}^{-k+1}=g(x).$ | (2) |

The substitution

$z:={y}^{-k+1}$ | (3) |

transforms (2) into

$$\frac{dz}{dx}+(-k+1)f(x)z=(-k+1)g(x)$$ |

which is a linear differential equation of first order. When one has obtained its general solution and made in this the substitution (3), then one has solved the Bernoulli equation (1).

## References

- 1 N. Piskunov: Diferentsiaal- ja integraalarvutus kõrgematele tehnilistele õppeasutustele. – Kirjastus Valgus, Tallinn (1966).

Title | Bernoulli equation |
---|---|

Canonical name | BernoulliEquation |

Date of creation | 2013-03-22 15:15:03 |

Last modified on | 2013-03-22 15:15:03 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 11 |

Author | pahio (2872) |

Entry type | Derivation |

Classification | msc 34C05 |

Synonym | Bernoulli differential equation |

Related topic | RiccatiEquation |