Second-order linear equations in two independent variables
In this chapter we classify the family of second-order linear equations for functions in two independent variables into three distinct types: hyperbolic (e.g., the wave equation), parabolic (e.g., the heat equation), and elliptic equations (e.g., the Laplace equation). It turns out that solutions of equations of the same type share many exclusive qualitative properties. We show that by a certain change of variables any equation of a particular type can be transformed into a canonical form which is associated with its type.
We concentrate in this chapter on second-order linear equations for functions in two independent variables x, y. Such an equation ...