This course explores analytic and numerical solution techniques for heat/mass diffusion and vibration/wave equations. Emphasis is placed on intuitive derivation of these equations, and analytic solution techniques like separation of variations, eigenfunction expansions, Fourier analysis, integral transforms, coordinate transforms, and special functions. Numerical solutions are introduced via finite difference methods. A key learning outcome of this course is understanding the central role that analytic solutions play in developing intuition about engineering physics, and how this is a fundamental step in learning to verify, validate, and properly use advanced computational modelling tools.