# A Universal Ordinary Differential Equation

## TL;DR

This paper proves the existence of a fixed polynomial ODE capable of approximating any positive continuous function with a unique solution, solving a longstanding open problem in differential equations.

## Contribution

It demonstrates that polynomial ODEs can be constructed to universally approximate functions with unique solutions, resolving Rubel's open problem from 1981.

## Key findings

- Existence of a fixed polynomial ODE approximating any positive continuous function.
- The solution to the ODE is necessarily analytic.
- The initial condition can be computed from the target function and error function.

## 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.