Within the Natural Sciences Tripos at Cambridge, in first and second year there is a dedicated mathematics subject (which was my favourite subject in both years).
The Cambridge course is very much a Mathematical Methods course, focussing on creative application rather than formality.
Unlike for physics, I haven’t followed the structure of the official course handout in these notes. Instead, I have chosen to group together certain topics (for example, Sturm-Liouville theory with the other ODE stuff).