We are interested in a formal method to solve first-order and second-order linear partial differential equations. Besides the solution itself, we must also pay attention to the initial conditions and the boundary conditions. Once the general solution is known and the initial conditions and boundary conditions are satisfied, we have the complete solution of a given problem. In usual problems, the initial and/or boundary conditions are given, and we must obtain the formal solution that satisfies the corresponding conditions. These solutions, here, are obtained by means of the method of separation of variables, a powerful tool to discuss linear differential equations. We introduce some basic concepts and the superposition principle. For the case in which we have more than two independent variables we present the method of separation of variables, with the corresponding boundary conditions. Some solved exercises are discussed step-by-step, among which the one dimensional wave equation. As an application we study the temperatures in a straight circular cylinder, a problem in which the Bessel functions appear naturally. We conclude with a list of proposed exercises; some interesting applications are left to the reader, some of them involving the d’Alembert equation, all of them with the corresponding answer and/or suggestion.