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)

References

[a1]	H.T. Davis, "Introduction to nonlinear differential and integral equations" , Dover, reprint (1962) pp. Chapt. II, §2