In this paper we study the notion of equivariant forms introduced in the authors' previous works. In particular, we completely classify all the equivariant forms for a subgroup of $\operatorname{SL}_2(\mathbb{Z})$ by means of the cross-ratio, the weight 2 modular forms, the quasimodular forms, as well as differential forms of a Riemann surface and sections of a canonical line bundle.

The purpose of this paper is to solve various differential equations having Eisenstein series as coefficients using various tools and techniques. The solutions are given in terms of modular forms, modular functions, and equivariant forms.