Consider the differential equation $P(y',y'',y''',y'''')=0$ on $\mathbb R$, where $P(x,y,z,w)$ is the homogeneous polynomial of degree $7$ given by $$ 3x^4yw^2-4x^4z^2w+6x^3y^2zw+24x^2y^4w-12x^3yz^3-29x^2y^3z^2+12y^7. $$ This example was given by Rubel in 1981 (Bulletin of the AMS), and he proved that for any continuous functions $f,g\colon\mathbb R\to\mathbb R$ with $g>0$ there is a solution $y$ of the differential equation satisfying $$ |y(t)-f(t)|\le g(t),\quad \text{for all } t\in\mathbb R. $$ Quite impressive. When one reads the proof one understands that all comes from the particular structure of the equation, but really impressive!

My question is the following:

Is there any polynomial of smaller degree leading to the same property?

A perhaps more ambitious question would be: what is the smallest degree of a polynomial having this property? In fact one could also vary the number of variables of the polynomial.

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    $\begingroup$ 3 years and no responses! Seems like it's ripe for mathoverflow, possibly a research grade question $\endgroup$ Jun 14, 2016 at 21:52
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    $\begingroup$ @frogeyedpeas More like 3 months? :) $\endgroup$
    – John B
    Jun 14, 2016 at 21:56

1 Answer 1


Perhaps it is paywalled, but this paper of C. Elsner claims to obtain a universal ODE out of a sixth-order polynomial equation. The title is "A Universal Differential Equation of Degree 6". The solutions of this equation are, like in Rubel's, $C^\infty$.

Wolfram mathworld describes two additional families (due to Duffin and Briggs) of universal (polynomial) differential equations, the solutions of which appear to be only $C^n$ for some $n$. These families have degree 3.


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