Newly updated by the author, this text explores the solution of partial differential equations by separating variables, rather than by conducting qualitative theoretical analyses of their properties. These qualitative features--uniqueness, existence, elegance of composition, and convergence modes--are substantiated by physical reasoning, rather than rigorous arguments. Geared toward applied mathematicians, physicists, engineers, and others seeking explicit solutions, the book offers heuristic justifications for each construction.The first three chapters review the necessary tools for understanding the separation of variables technique: basics of ordinary differential equations, Frobenius-series construction and properties of Bessel functions, and Fourier analysis. Subsequent chapters explore the exposition of the algorithmic nature of the separation of variables process, based on a sequence of steps that infallibly leads to the solution expansion, regardless of the nature of the boundary conditions.