The equation is parabolic at a point when $\boxed{b^2-ac=0}$ and at least one of $a,b,c$ is nonzero. The principal <matrix> then has rank one, with a repeated <characteristic direction>. If all three vanish, there is no second-order <principal part of a meromorphic function>[principal part] there and the usual parabolic classification does not apply.

Solution

Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2009 / Differential equations / 8 / ii / Solution

 Solution (source code)