This equation can be cast into the form of the hypergeometric differential equation. It has two linearly independent solutions, called the periods of elliptic functions. The ratio of the two periods is equal to the period ratio τ, the standard coordinate on the upper-half plane. However, the ratio of two solutions of the hypergeometric equation is also known as a Schwarz triangle map.
J. Harnad and J. McKay, Modular solutions to equations of generalized Halphen type, Proc. R. Soc. London A 456 (2000), 261–294,
(Provides a readable introduction, some history, references, and various interesting identities and relations between solutions)
J. Harnad, Picard–Fuchs Equations, Hauptmoduls and Integrable Systems, Chapter 8 (Pgs. 137–152) of Integrability: The Seiberg–Witten and Witham Equation (Eds. H.W. Braden and I.M. Krichever, Gordon and Breach, Amsterdam (2000)).
(Provides further examples of Picard–Fuchs equations satisfied by modular functions of genus 0, including non-triangular ones, and introduces Inhomogeneous Picard–Fuchs equations as special solutions to isomonodromic deformation equations of Painlevé type.)