Discussion
The solution of \( \dfrac{\partial^{2}z}{\partial x^{2}} - \dfrac{\partial^{2}z}{\partial y^{2}} = 0 \) is
Answer: D
With \( D' = mD \) the auxiliary equation is
\[ 1 - m^{2} \;=\; 0 \quad \Longrightarrow \quad m \;=\; \pm 1, \]
so the two roots give \( z = f(y + x) + g(y - x) \). Imaginary roots \( m = \pm i \) would come from the *sum* of the two second derivatives, which is Laplace's equation
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