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