= Solution
The principal coefficients are $A=-xy$, $B=0$, $C=1$, in the convention $Au_{xx}+2Bu_{xy}+Cu_{yy}$. Their <discriminant> is \b[$B^2-AC=xy$]. Thus it is a <hyperbolic partial differential equation> where $xy>0$ and an <elliptic partial differential equation> where $xy<0$. Its principal part has <parabolic type> where $xy=0$; however the full equation contains $u_y/y$ and is undefined on $y=0$. On its actual domain $y\ne0$, the parabolic locus is $x=0$.
The <characteristic equation of a partial differential equation> is
$$
\boxed{A(dy)^2-2B\,dx\,dy+C(dx)^2=0,\qquad (dx)^2-xy(dy)^2=0.}
$$
In a hyperbolic region this gives two distinct real <characteristic directions>. The singular lower-order coefficient should not be mistaken for an ordinary parabolic point of the full equation on $y=0$.
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