= Characteristic equation of a partial differential equation
{title2=$A(dy)^2-2B\,dx\,dy+C(dx)^2=0$}
For principal part $Au_{xx}+2Bu_{xy}+Cu_{yy}$, 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.
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