The principal <matrix> is $\begin{pmatrix}a&b\\b&c\end{pmatrix}$. The equation is elliptic at a point when $\boxed{b^2-ac<0}$, equivalently the principal form has <eigenvalues> of the same nonzero sign and no real <characteristic directions>.

Solution

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

 Solution (source code)