On level 2 modular differential equations

In this paper, we explore the modular differential equation [Formula: see text] on the upper half-plane [Formula: see text], where F is a weight 4 modular form for [Formula: see text]. Our approach centers on solving the associated Schwarzian equation [Formula: see text], where [Formula: see text] represents the Schwarzian derivative of a meromorphic function h on [Formula: see text]. We derive conditions under which the solutions to this equation are modular functions for subgroups of the modular group and provide explicit expressions for these solutions in terms of classical modular functions. Key tools in our analysis include the theory of equivariant functions on the upper half-plane and the representation theory of level 2 subgroups of the modular group. The corresponding case where F is a modular form for the full modular group [Formula: see text] has been previously addressed in previous work by the second author.