The Infinities in the Non-Local Existence Problem of Ordinary Differential Equations

Abstract

The standard general existence theorem for the problem of initial values assigned for systems of ordinary differential equations is strictly local in nature. There seem to be just two standard types which are particular enough to allow a control of the solutions in the large. The first of these classical cases is represented by systems of linear differential equations, the second by those systems the solutions of which depend on inversions of quadratures. This latter type is exemplified, in the complex domain, by the differential equation dp/dz = (p3 + ap + b) I of the elliptic function p (z) and, in the real domain, by Liouville's separable conservative systems with ru degrees of freedom. In fact, if x denotes any of the n separating coordinates, the energy relation of each degree of freedom is a differential equation of the form dx/dt = f(x). As illustrated by the example f(x) = X113, where -oo < < oo, the mere continuity of f(x) is insufficient for the uniqueness of the