Classification of second order PDEs

A linear, second order PDE with two independent variables $x,y$ and constant coefficients $a_{xx} \in \mathbb{R}$, $a_{xx}\ne 0$ has the form:

$$a_{11}u_{xx}+2a_{12}u_{xy}+a_{22}u_{yy}+a_1u_x+a_2u_y+a_0u\equiv0.$$ (14)

1. Elliptic type
   $a_{12}^2 < a_{11}a_{22}$
   
   (14) $\rightarrow$ $u_{xx}+u_{yy}+\ldots\equiv0$. (e.g. Laplace equation)

2. Hyperbolic type
   $a_{12}^2=a_{11}a_{22}$ and $(a_{11},a_{12},a_{22})\ne (0,0,0)$
   
   (14) $\rightarrow$ $u_{xx}+\ldots\equiv0$. (e.g. Wave equation)

3. Parabolic type
   $a_{12}^2 > a_{11}a_{22}$
   
   (14) $\rightarrow$ $u_{xx}-u_{yy}+\ldots\equiv0$. (e.g. Diffusion equation)

``$\ldots$'' represents the terms of order $\le 1$.