The general second-order linear partial differential equation gives a standard form for many important equations that arise in mathematical physics. In this page, we describe the structure of such an equation in $n$ independent variables, identify the roles of its coefficient functions, and explain the distinction between homogeneous and nonhomogeneous equations.

Second-order linear partial differential equations occur frequently in mathematical models involving heat flow, wave motion, potential theory, and other physical processes. At this introductory stage, our purpose is not to solve these equations, but to understand their general algebraic form and the meaning of each term appearing in that form.