  The main goal of this article is to compute the class of the divisor of
$\overline{\mathcal{M}}_3$ obtained by taking the closure of the image of
$\Omega\mathcal{M}_3(6;-2)$ by the forgetful map. This is done using Porteous
formula and the theory of test curves. For this purpose, we study the locus of
meromorphic differentials of the second kind, computing the dimension of the
map of these loci to $\mathcal{M}_g$ and solving some enumerative problems
involving such differentials in low genus. A key tool of the proof is the
compactification of strata recently introduced by
Bainbridge-Chen-Gendron-Grushevsky-M\"oller.
