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_18: 
@cite_14: I construct a new universal differential equation (B), in the sense of Rubel. That is, its solutions approximate to arbitrary accuracy any continuous function on any interval of the real line.
@cite_8: Abstract Fourth-order differential equations such as 16y′my′2 - 32ymyny′ + 17y03 = 0 are developed. It is shown that the equation is “universal” in the sense that any continuous function can be approximated with arbitrary accuracy over the whole x axis by a solution y(x) of the equation. This solution is a piecewise polynomial of degree 9 and of class C4.
First, let us mention that Rubel's universal differential equation has been extended in several papers. In particular, Duffin proved in @cite_8 that implicit universal differential equations with simpler expressions exists, such as @math for any @math The idea of @cite_8 is basically to replace the @math function @math of Rub81 by some piecewise polynomial of fixed degree, that is to say by splines. Duffin also proves that considering trigonometric polynomials for function @math leads to the universal differential equation @math This is done at the price of approximating function @math respectively by splines or trigonometric splines solutions which are @math (and @math can be taken arbitrary big) but not @math as in @cite_18 . Article @cite_14 proposes another universal differential equation whose construction is based on Jacobian elliptic functions. Notice that @cite_14 is also correcting some statements of @cite_8 .