Discussion
The partial differential equation
\[ \begin{aligned} &x^{2} \frac{\partial^{2} z}{\partial x^{2}} - y^{2} \frac{\partial^{2} z}{\partial y^{2}} \\ &= xyz \end{aligned} \]
is :
Answer: A
The equation is hyperbolic.
For \(A z_{xx} + 2B z_{xy} + C z_{yy} + \ldots = 0\) the type is set by the discriminant \(B^{2} - AC\). Here
\[ \begin{aligned} A &= x^{2},\quad B = 0,\quad C = -y^{2} \\ B^{2} - AC &= x^{2}y^{2} \end{aligned} \]
which is positive whenever \(xy \neq 0\), so the equation is hyperbolic away from the coordinate axes (where it degenerates to parabolic). The right-hand side \(xyz\) involves no second derivatives and does not affect the type.
It would be elliptic only if \(A\) and \(C\) had the same sign, and parabolic only if the discriminant vanished identically. 'Laplacian' is not a type; the Laplace equation is the model elliptic equation.
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