= Solution
For a nonzero principal part the <discriminant> $\Delta=b^2-ac$ classifies the equation: it is \b[hyperbolic for $\Delta>0$, parabolic for $\Delta=0$, and elliptic for $\Delta<0$]. These are the three definitions requested by the inline roman labels.
In the hyperbolic case the <characteristic equation> $a(dy)^2-2b\,dx\,dy+c(dx)^2=0$ gives two distinct real tangent directions. Locally, with suitably smooth coefficients, solve the corresponding characteristic <ordinary differential equations> to obtain two independent families of curves. Label their level sets $\phi=\text{constant}$ and $\psi=\text{constant}$. Their gradients satisfy
$$
a\phi_x^2+2b\phi_x\phi_y+c\phi_y^2=0,\qquad a\psi_x^2+2b\psi_x\psi_y+c\psi_y^2=0,
$$
and the coordinate Jacobian is nonzero. The <chain rule> therefore eliminates $u_{\phi\phi}$ and $u_{\psi\psi}$. The remaining mixed coefficient $2[a\phi_x\psi_x+b(\phi_x\psi_y+\phi_y\psi_x)+c\phi_y\psi_y]$ is nonzero: otherwise the invertible coordinate change would send the nondegenerate principal quadratic form to <zero>. Divide by this coefficient and absorb all first-order terms into $g$, giving $u_{\phi\psi}=g(\phi,\psi,u,u_\phi,u_\psi)$. This canonical reduction is local; a global chart is not automatic.
For the specific equation $a=y,b=0,c=-1$, so it is hyperbolic exactly on \b[$y>0$], elliptic for $y<0$, and degenerate parabolic on $y=0$. For $\phi=3x-2y^{3/2}$ and $\psi=3x+2y^{3/2}$, the Jacobian is $18\sqrt y\ne0$. Direct differentiation gives
$$
yu_{xx}-u_{yy}=36y\,u_{\phi\psi}+\frac3{2\sqrt y}(u_\phi-u_\psi).
$$
Since $\psi-\phi=4y^{3/2}$, the canonical form is
$$
\boxed{u_{\phi\psi}=\frac{u_\psi-u_\phi}{6(\psi-\phi)}.}
$$
The characteristic families are $3x-2y^{3/2}=c$ and $3x+2y^{3/2}=c$, with slopes $dx/dy=\pm\sqrt y$. They are distinct for $y>0$ and meet the degeneracy line with coincident vertical tangent directions.
\Image[/past-exam-of-the-algebra-analysis-and-differential-equations-2-course-of-the-university-of-oxford/2005-characteristics.png]
{title=The two characteristic families in the hyperbolic region}
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