= Pointwise classification of a second-order partial differential equation
{title2=$\Delta=B^2-AC$}
For $Au_{xx}+2Bu_{xy}+Cu_{yy}+\cdots=0$, the sign of $\Delta=B^2-AC$ distinguishes a <hyperbolic partial differential equation> when $\Delta>0$, an <elliptic partial differential equation> when $\Delta<0$, and a <parabolic partial differential equation> when $\Delta=0$ and the <principal symbol> has rank one. If $A=B=C=0$, the second-order <principal symbol> has rank zero and the differential order drops; this fully degenerate case must be checked separately for compatibility with the lower-order terms and forcing.
Back to article page
