There are several problems in which the equation to be solved involves more than one independent variable, for example, the differential equation describing the motion of a vibrating spring. We discuss first and second order linear partial differential equations with one dependent variable and two independent variables. We introduce a classification for a linear second order partial differential equation with one dependent variable and only two independent variables; three types are discussed, parabolic equation, hyperbolic equation and elliptic equation. The canonical form for these three cases are presented and as a particular case we discuss the general linear second order partial differential equation with constant coefficients. Some solved exercises are discussed step-by-step, e.g. exercises involving the classification and reduction to the canonical form. As an application we study the two-dimensional Laplace equation written in Cartesian coordinates and in polar coordinates. We conclude with a list of proposed exercises, among them some interesting applications are left to the reader; one of them is the Tricomi equation, which has non constant coefficients; all of them with the answer and/or suggestion.