Modular differential equations of second order with regular singularities at elliptic points for $SL_2(\mathbb{Z})$

We give a definition of the modular differential equations of weight $k$ for a discrete subgroup for $\Gamma \subset SL_2(\mathbb{R})$; in this paper we set $\Gamma = SL_2(\mathbb{Z})$. We solve such equations admitting regular singularities at elliptic points for $SL_2(\mathbb{Z})$ in terms of the Eisenstein series and the Gauss hypergeometric series. Furthermore, we give a series of such modular differential equations parametrized by an even integer $k$, and discuss some properties of solution spaces. We find several equations which are solved by a modular form of weight $k$.