# A universal differential equation

• Published in 1981
In the collections
There exists a non trivial fourth-order algebraic differential equation $P(y',y'',y''',y'''') = 0,$ where $P$ is a polynomial in four variables, with integer coefficients, such that for any continuous function $\phi$ on $(-\infty,\infty)$ and for any positive continuous function $\varepsilon(t)$ on $(-\infty,\infty)$, there exists a $C^{\infty}$ solution $y$ of $P=0$ such that $|y(t)-\phi(t)| \lt \varepsilon(t)\) \forall t \in (-\infty,\infty)$

### BibTeX entry

@article{Auniversaldifferentialequation,
title = {A universal differential equation},
abstract = {There exists a non trivial fourth-order algebraic differential equation $P(y',y'',y''',y'''') = 0,$ where $P$ is a polynomial in four variables, with integer coefficients, such that for any continuous function $\phi$ on $(-\infty,\infty)$ and for any positive continuous function $\varepsilon(t)$ on $(-\infty,\infty)$, there exists a $C^{\{}\infty{\}}$ solution $y$ of $P=0$ such that $|y(t)-\phi(t)| \lt \varepsilon(t)\) \forall t \in (-\infty,\infty)$},
url = {https://projecteuclid.org/euclid.bams/1183548125},
year = 1981,
author = {Lee A. Rubel},
comment = {},
urldate = {2019-05-09},
collections = {fun-maths-facts,unusual-computers}
}