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# isocline

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

If the family $\Gamma$ has the differential equation

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

then the equation of any isocline of $\Gamma$ has the form

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

where $K$ is constant.

For example, the family

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

of exponential curves satisfies the differential equation $\frac{dy}{dx}=Ce^{{Cx}}$ or $\frac{dy}{dx}=Cy$, whence the isoclines are $Cy=K$, i.e. they are horizontal lines.

Related:

OrthogonalCurves

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## Mathematics Subject Classification

53A25*no label found*53A04

*no label found*51N05

*no label found*

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