  Let $C \langle \boldsymbol{t} \rangle$ be the differential field generated by
$l$ differential indeterminates $\boldsymbol{t}=(t_1, \dots, t_l)$ over an
algebraically closed field $C$ of characteristic zero. In this article we
present an explicit linear parameter differential equation over $C \langle
\boldsymbol{t} \rangle$ with differential Galois group $\mathrm{SL}_{l+1}(C)$
and show that it is a generic equation in the following sense: If $F$ is an
algebraically closed differential field with constants $C$ and $E/F$ is a
Picard-Vessiot extension with differential Galois group $H(C) \subseteq
\mathrm{SL}_{l+1}(C)$, then a specialization of our equation defines a
Picard-Vessiot extension differentially isomorphic to $E/F$.
