Characteristic equation of a partial differential equation ($A(dy{)}^2-2Bdxdy+C(dx{)}^2=0$)

For principal part $A{u}_{xx}+2B{u}_{xy}+C{u}_{yy}$, the displayed quadratic relation determines the characteristic directions. Curves tangent to them are characteristic curves. Lower-order terms do not enter this relation, but they may be singular where the principal coefficients alone suggest an otherwise ordinary classification. In particular, classifying a principal part on a singular line does not extend the full differential equation onto that line.