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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  $\Gamma=\Gamma_0^+(N)$ generated by $\Gamma_0(N)$ and the  Atkin-Lehner involutions for $N=1,2,3$  ($\Gamma_0^+(1)=\mathrm{SL}(2,\mathbb Z)$). Firstly, we note that aquasimodular  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  $\Gamma_0^+(N)$, $N=1,2,3$.

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