  We consider the spaces $\mathcal{F}_\mu$ of polynomial $\mu$-densities on the
line as $\mathfrak{sl}(2)$-modules and then we compute the cohomological spaces
$\mathrm{H}^2_\mathrm{diff}(\mathfrak{sl}(2),
\mathcal{D}_{\bar{\lambda},\mu})$, where $\mu\in \mathbb{R}$,
$\bar{\lambda}=(\lambda_1,\dots,\lambda_n)
  \in\mathbb{R}^n$ and $\mathcal{D}_{\bar{\lambda},\mu}$ is the space of
$n$-ary differential operators from
$\mathcal{F}_{\lambda_1}\otimes\cdots\otimes
  \mathcal{F}_{\lambda_n}$ to $\mathcal{F}_\mu$.
