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We consider the spaces $\mathcal{F}_\mu$ of polynomial $\mu$-densities on theline 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 $\mathcal{F}_\mu$.

Second cohomology space of $\frak {sl}(2)$ acting on the space of $n$-ary differential operators on $\mathbb{R}$