= Solution
{c}
In the convention $Av_{xx}+2Bv_{xy}+Cv_{yy}=\cdots$, the coefficients of the <principal symbol> are
$$
A=x,\qquad B=y+\frac{\beta x}{2},\qquad C=0,\qquad B^2-AC=\frac{(2y+\beta x)^2}{4}.
$$
Thus the <second-order partial differential equation> is
$$
\boxed{\begin{array}{ll}
\text{hyperbolic},&2y+\beta x\ne0,\\
\text{parabolic},&2y+\beta x=0,\quad x\ne0.
\end{array}}
$$
There 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.
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