# isocline

Let $\mathrm{\Gamma}$ be a family of plane curves. The isocline of $\mathrm{\Gamma}$ is the locus of the points, in which all members of $\mathrm{\Gamma}$ have an equal slope.

If the family $\mathrm{\Gamma}$ has the differential equation^{}

$$F(x,y,\frac{dy}{dx})=0,$$ |

then the equation of any isocline of $\mathrm{\Gamma}$ has the form

$$F(x,y,K)=0$$ |

where $K$ is .

For example, the family

$$y={e}^{Cx}$$ |

of exponential^{} (http://planetmath.org/ExponentialFunction) curves satisfies the differential equation $\frac{dy}{dx}=C{e}^{Cx}$ or $\frac{dy}{dx}=Cy$, whence the isoclines are $Cy=K$, i.e. they are horizontal lines.

http://en.wikibooks.org/wiki/Differential_Equations/Isoclines_1Wiki

Title | isocline |
---|---|

Canonical name | Isocline |

Date of creation | 2013-03-22 18:05:52 |

Last modified on | 2013-03-22 18:05:52 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 6 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 53A25 |

Classification | msc 53A04 |

Classification | msc 51N05 |

Related topic | OrthogonalCurves |