Differential polynomial identities of upper triangular matrices of size three

We determine the differential polynomial identities of $3\times 3$ upper triangular matrices over a base field of characteristic zero, under the action of its full Lie algebra of derivations. We compute the exact differential codimension sequence of the multilinear ones and describe their $S_n$-structure by means of an explicit decomposition of the $S_n$-cocharacter of their proper part.