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a set of points in the xy-plane such that at each point a certain direction is defined. The direction is usually represented by an arrow passing through the point.
If the equation yʹ = f(x, y) is given, then at each point (x0, y0) of some region in the xy-plane the value of the slope k = f (x0, y0) of the tangent to the integral curve passing through this point is known, so that the direction of the tangent can be represented by an arrow. Thus the differential equation defines a direction field; conversely, a direction field in some region of the xy-plane defines a differential equation of the form yʹ = f(x, y). By sketching sufficiently many isoclines (curves joining points at which the direction field f (x, y) = C, C a constant, has the same value), we can obtain approximations to the integral curves, that is, curves having the prescribed directions as tangents (the isocline method).
Thedirection field of the equation yʹ = x2 + y2 is shown in Figure 1. The light curves (circles) represent isoclines, and the thicker curves represent integral curves.
REFERENCESStepanov, V. V. Kurs differentsial’nykh uravnenii, 8th ed. Moscow, 1959.
Petrovskii, I. G. Lektsii po teorii obyknovennykh differentsial’nykh uravnenii, 6th ed. Moscow, 1970.