CODEE Expects that every Mathematician will do their ODEs: From the Battle of Trafalgar to Calculus (or Nelson to Newton)

Using the historical Battle of Trafalgar as a case study, we introduce students to mathematical modeling to describe and explore real-world systems. Starting with the question—how do force sizes and combat effectiveness determine which side wins?—we show how to use a blend of mathematics and computational tools to investigate the answer. Beginning by defining a simple differential equation model for combat dynamics, we analyze its behavior using differential equations and accessible algebraic techniques. We show how computational methods and simulations can support systematic, visual exploration and suggest conjectures that mathematical analysis can then establish. This material could serve as the foundation of a lesson plan which integrates historical background, computational tools, and mathematical theory to motivate students and foster an interest in using mathematical modeling to solve complex, real-world problems.