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of a first-order differential equation

A set of points in the -plane at which the inclinations of the direction field defined by equation

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

is the set

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

going through the point is . For example, the -isocline is defined by the equation and consists of just those points of the -plane at which the solutions of equation

have horizontal tangents. The -isocline of

is simultaneously a solution of

if and only if it is a line with slope .

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 , and if the corresponding inclinations of the integral curves are drawn (the method of isoclines). It is also useful to construct the -isocline, defined by the equation ; at the points of the -isocline the integral curves of equation

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

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

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

the -isocline is defined as the set

In the case of a second-order autonomous system,

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


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



[a1] H.T. Davis, "Introduction to nonlinear differential and integral equations" , Dover, reprint (1962) pp. Chapt. II, §2
How to Cite This Entry:
Isocline. N.Kh. Rozov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098