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In the convention $A{v}_{xx}+2B{v}_{xy}+C{v}_{yy}=\cdots$, the coefficients of the principal symbol areThus the second-order partial differential equation isThere is no region of elliptic partial differential equation type. The terms hyperbolic, parabolic and elliptic refer to the pointwise classification of a second-order partial differential equation. At $(0,0)$ every second-order coefficient vanishes, so the principal symbol has rank zero and the equation is degenerate rather than genuinely parabolic there. Moreover the right side is $2$, so no twice continuously differentiable classical solution can satisfy the equation at the origin.

Along a curve $s\mapsto (x(s),y(s))$, the chain rule givesTogether with the second-order partial differential equation, these form a linear system for $(u_{xx},u_{xy},u_{yy})$ whose coefficient matrix isA characteristic curve is precisely one along which this system is singular, equivalently where the principal symbol vanishes on a normal covector. Taking the determinant yields