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)\]

@article{Auniversaldifferentialequation,
key = {Auniversaldifferentialequation},
type = {article},
title = {A universal differential equation},
author = {Lee A. Rubel},
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)\]},
comment = {},
date_added = {2019-05-09},
date_published = {1981-09-14},
urls = {https://projecteuclid.org/euclid.bams/1183548125},
collections = {Fun maths facts,Unusual computers},
url = {https://projecteuclid.org/euclid.bams/1183548125},
year = 1981,
urldate = {2019-05-09}
}