A Universal Ordinary Differential Equation

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 varphi on the reals, and for any positive continuous function ε(t), it has a C^∞ solution with | y(t) - varphi(t) | < ε(t) for all t. 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 never has a unique solution, even with a finite number of conditions of the form y(ki)(ai)=bi. The question whether one can require the solution that approximates varphi 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 fixed polynomial ODE such that for any varphi and ε(t) there exists some initial condition that yields a solution that is ε-close to varphi 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.