Abstract
In this paper, we investigate the non-modular solutions to the Schwarz differential equation {f, tau} = sE(4)(tau) where E-4(tau) is the weight 4 Eisenstein series and s is a complex parameter. In particular, we provide explicit solutions for each s = 2 pi(2)(n/6)(2) with n equivalent to 1 mod 12. These solutions are obtained as integrals of meromorphic weight 2 modular forms. As a consequence, we find explicit solutions to the differential equation y '' + pi(2)n(2)/36E(4)y = 0 for each n equivalent to 1 mod 12 generalizing the work of Hurwitz and Klein on the case n = 1. Our investigation relies on the theory of equivariant functions on the complex upper half-plane. This paper supplements a previous work where we determine all the parameters s for which the above Schwarzian equation has a modular solution.