Discussion
The partial differential equation
\[ \begin{aligned} &\frac{\partial^{2} z}{\partial x^{2}} + y\frac{\partial^{2} z}{\partial y^{2}} = 0, \\ &y \gt 0 \end{aligned} \]
is :
Answer: C
The equation is elliptic.
Compare with \(Az_{xx} + Bz_{xy} + Cz_{yy} = 0\): here \(A = 1\), \(B = 0\), \(C = y\). Then
\[ B^{2} - 4AC = -4y \lt 0 \quad (y \gt 0) \]
A negative discriminant means elliptic. It would be hyperbolic for \(y \lt 0\) and parabolic on \(y = 0\), but the question restricts to \(y \gt 0\). 'Laplacian' is not a type; Laplace's equation is itself the model elliptic equation.
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