An astonishing fact was established by Lee A. Rubel (1981): there exists a fixed non-trivial fourth-order polynomial differential algebraic equation (DAE) such that for any positive continuous function @math on the reals, and for any positive continuous function @math , it has a @math solution with @math for all @math . Lee A. Rubel provided an explicit example of such a polynomial DAE. Other examples of universal DAE have later been proposed by other authors. However, Rubel's DAE has a unique solution, even with a finite number of conditions of the form @math . The question whether one can require the solution that approximates @math to be the unique solution for a given initial data is a well known open problem [Rubel 1981, page 2], [Boshernitzan 1986, Conjecture 6.2]. In this article, we solve it and show that Rubel's statement holds for polynomial ordinary differential equations (ODEs), and since polynomial ODEs have a unique solution given an initial data, this positively answers Rubel's open problem. More precisely, we show that there exists a polynomial ODE such that for any @math and @math there exists some initial condition that yields a solution that is @math -close to @math at all times. In particular, the solution to the ODE is necessarily analytic, and we show that the initial condition is computable from the target function and error function.
@cite_9: Etude des solutions des equations differentielles algebriques en relation avec l'approximation uniforme des fonctions
@cite_13: Abstract A universal ordinary differential equation C ∞ of order 3 is constructed here. The equation is universal in the sense that any continuous function on a real segment can be approximated by a solution of this equation with an arbitrary accuracy in uniform norm.
All the results mentioned so far are concerned with approximations of continuous functions over the whole real line. Approximating functions over a compact domain seems to be a different (and somewhat easier for our concerns) problem, since basically by compactness, one just needs to approximate the function locally on a finite number of intervals. A 1986 reference survey discussing both approximation over the real line and over compacts is @cite_9 . Recently, over compact domains, the existence of universal ordinary differential equation @math of order @math has been established in @cite_13 : it is shown that for any @math , there exists a third order @math differential equation @math whose solutions are dense in @math . Notice that this is not obtained by explicitly stating such an order @math universal ordinary differential, and that this is a weaker notion of universality as solutions are only assumed to be arbitrary close over a compact domain and not all the real line. Order @math is argued to be a lower bound for Lipschitzian universal ODEs @cite_13 .