The Serre derivative is a differential operator that raises the weight of a modular form by 2. One possible generalization of the Serre derivative to the setting of Jacobi forms is a modification of the heat operator involving the quasi-modular Eisenstein series E_2. In this talk, I will present an approach to constructing modular differential equations for Jacobi forms with respect to this operator. This method makes it possible to describe solutions of first- and second-order modular differential equations (Kaneko–Zagier type equations), to construct higher-order differential equations, and to obtain applications to the elliptic genus of Calabi–Yau manifolds. This is joint work with Valery Gritsenko.