Title: A Universal Ordinary Differential Equation,
Abstract: 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 $\epsilon(t)$, it has a $\mathcal{C}^\infty$
solution with $| y(t) - \varphi(t) | < \epsilon(t)$ 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
\emph{never} has a unique solution, even with a finite number of conditions of
the form $y^{(k_i)}(a_i)=b_i$.
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 \textbf{fixed} polynomial ODE such that
for any $\varphi$ and $\epsilon(t)$ there exists some initial condition that
yields a solution that is $\epsilon$-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.