A Note on First Order Differential Equations of Degree Greater than One and Wavefront Evolution

by the implicit function theorem. This second form is far more convenient than the first (for example one can find approximate solutions using the relationship Sy = f(x, y)dx (<5x small), or in the analytic case find a series solution by repeated differentiation). Let us assume that 0 is a regular value of the function F, so F~ (0) is a smooth surface in IR. The locus F = Fp — 0 is the set of critical points of the projection n : F 1 (0) -* U to the (x, y) plane and its image under n is the apparent contour of F 1 (0) in the p-direction. One problem of some interest is to describe the nature of the integral curves of the differential equation (1) near this apparent contour; see [6, 8]. Using the standard Legendre transformation we now show how this problem is related to families of curves and their duals, and hence to wavefront evolution. If we apply the Legendre transformation ([7, p. 68]) X = p, Y = xp — y,P = x to our surface we obtain a new surface G{X, Y, P) = F(P, XPY, X) = 0 in U. (Recall that the Legendre transformation is a smooth involution.) If GP =f=0 at the point (Xo, Yo, Po) corresponding to (x0, y0, p0) we can write the