  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 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
  $\Gamma_0^+(N)$, $N=1,2,3$.
