A differential equation satisfied by the arithmetic Kodaira–Spencer class

The arithmetic Kodaira–Spencer class of the universal elliptic curve was introduced in [A. Buium, Differential modular forms, J. Reine Angew. Math. 520 (2000) 95–167]; its reduction mod p was explicitly computed by Hurlburt [C. Hurlburt, Isogeny covariant differential modular forms modulo p, Compos. Math. 128 (1) (2001) 17–34]. In this paper the complicated expression of Hurlburt is shown to be the unique solution of a simple partial differential equation subject to a certain initial condition and weight condition.