  Let $V$ be a strongly regular vertex operator algebra and let $\frak{ch}_V$
be the space spanned by the characters of the irreducible $V$-modules.\ It is
known that $\frak{ch}_V$ is the space of solutions of a so-called \emph{modular
linear differential equation (MLDE)}.\ In this paper we obtain a
near-classification of those $V$ for which the corresponding MLDE is
irreducible and monic of order $2$.\ As a consequence we derive the complete
classification when $V$ has exactly two simple modules.\ It turns out that $V$
is either one of four affine Kac-Moody algebras of level $1$, or the Yang-Lee
Virasoro model of central charge ${-}22/5$.\ Our proof establishes new
connections between the characters of $V$ and Gauss hypergeometric series, and
puts the finishing touches to work of Mathur, Mukhi and Sen who first
considered this problem forty years ago.
