# Isocline

of a first-order differential equation

$$\tag{* } y ^ \prime = f ( x, y)$$

A set of points in the $( x, y)$- plane at which the inclinations of the direction field defined by equation

are one and the same. If $k$ is an arbitrary real number, then the $k$- isocline of equation

is the set

$$\{ {( x, y) } : {f ( x, y) = k } \}$$

(in general, this is a curve); at each of its points the (oriented) angle between the $x$- axis and the tangent to the solution of

going through the point is ${ \mathop{\rm arc} \mathop{\rm tan} } k$. For example, the $0$- isocline is defined by the equation $f ( x, y) = 0$ and consists of just those points of the $( x, y)$- plane at which the solutions of equation

have horizontal tangents. The $k$- isocline of

is simultaneously a solution of

if and only if it is a line with slope $k$.

A rough qualitative representation of the behaviour of the integral curves (cf. Integral curve) of

can be obtained if the isoclines of the given equation are constructed for a sufficiently frequent choice of the parameter $k$, and if the corresponding inclinations of the integral curves are drawn (the method of isoclines). It is also useful to construct the $\infty$- isocline, defined by the equation $1/ {f ( x, y) } = 0$; at the points of the $\infty$- isocline the integral curves of equation

have vertical tangents. The (local) extreme points of the solutions of

can lie on the $0$- isocline only, and the points of inflection of the solution can lie only on the curve

$$\frac{\partial f ( x, y) }{\partial x } + f ( x, y) \frac{\partial f ( x, y) }{\partial y } = 0.$$

For a first-order equation not solvable with respect to the derivative,

$$F ( x, y, y ^ \prime ) = 0,$$

the $k$- isocline is defined as the set

$$\{ {( x, y) } : {F ( x, y, k) = 0 } \} .$$

In the case of a second-order autonomous system,

$$\dot{x} = f ( x, y),\ \ \dot{y} = g ( x, y),$$

the set of points in the phase plane at which the vectors of the phase velocity are collinear is an isocline of the equation

$$\frac{dy }{dx } = \ \frac{g ( x, y) }{f ( x, y) } .$$

#### References

 [1] W.W. [V.V. Stepanov] Stepanow, "Lehrbuch der Differentialgleichungen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)