Equivariant solutions to modular Schwarzian equations

For every positive integer r, we solve the modular Schwarzian differential equation {h,τ}=2π2r2E4, where E4 is weight 4 Eisenstein series, by means of equivariant functions on upper half-plane. This paper supplements previous works authors [20], [21], same has been solved for infinite families rational values r. also leads to solutions y″+r2π2E4y=0 These are quasi-modular forms SL2(Z) if r even or subgroup index 2, SL2(Z)2, odd.