# Definition(s) > [!NOTE] Definition (General Form of $2$nd order linear PDE) > A [[Partial differential equations|PDE]] of the form $A\partial_{xx}u + 2B\partial_{xt}u+C\partial_{tt}u+G(u,\partial_{x}u,\partial_{t}u)=0,\tag{*}$where $A,B,C$ are constants. > > We define its principal (or leading) part of the equation is $A\partial_{xx}u+B\partial_{xt}u+C\partial_{tt}u.$ > > We define its discriminant as $D=B^2 -4ac.$ > [!Example] Example > Contents # Properties(s) **Classification**: Equation $(*)$ is called [[Hyperbolic partial differential equations|hyperbolic]] if $D>0$; [[Elliptic partial differential equation|ellliptic]] iff $D<0$ and [[Parabolic partial differential equations|parabolic]] if $D=0.$ # Application(s) **More examples**: # Bibliography