Second-order algebraic methods provide a natural and expressive formal framework in which to develop correct computing systems. In this paper we consider using second-order algebraic methods to specify real-time systems and to verify their associated safety and utility properties. We demonstrate our ideas by presenting a detailed case study of the railroad crossing controller, a benchmark example in the real-time systems community. This case study demonstrates how real-time constraints can be modelled naturally using second-order algebras and illustrates the substantial expressive power of second-order equations.