In mathematical physics one attempts, starting from the basic equations and assumptions of physics (such as Newton's equation, the Boltzmann distribution or the Schrödinger equation) to derive physical facts mathematically.
So the focus is the physical problem (eg the question of the stability of the solar system, the reason for the existence of crystals or the localization of electrons in amorphous solids).

The mathematical methods used in the solution of the given problem are mainly  analyticals or geometrical, but algebraic techniques also play a role.

Roughly speaking

• classical mechanics corresponds to the theory of ordinary differential equations, 
• (classical) statistical mechanics to probability theory, and 
• quantum mechanics to functional analysis.