Quasimodular forms and modular differential equations which are not apparent at cusps: I

In this paper, we explore a two-way connection between quasimodular forms of depth $1$ and a class of second-order modular differential equations with regular singularities on the upper half-plane and the cusps. Here we consider the cases $\mathrm{\Gamma }={\mathrm{\Gamma }}_0^{+}(N)$ generated by ${\mathrm{\Gamma }}_0(N)$ and the Atkin-Lehner involutions for $N=1,2,3$ (${\mathrm{\Gamma }}_0^{+}(1)=\mathrm{S}\mathrm{L}(2,\mathbb{Z})$). Firstly, we note that a quasimodular form of depth $1$, after divided by some modular form with the same weight, is a solution of a modular differential equation. Our main results are the converse of the above statement for the groups ${\mathrm{\Gamma }}_0^{+}(N)$, $N=1,2,3$.