# 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 .

For example, the family

 $y=e^{Cx}$

of exponential (http://planetmath.org/ExponentialFunction) 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.

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

Title isocline Isocline 2013-03-22 18:05:52 2013-03-22 18:05:52 pahio (2872) pahio (2872) 6 pahio (2872) Definition msc 53A25 msc 53A04 msc 51N05 OrthogonalCurves