Discussion
The solution of the partial differential equation :
\(\dfrac{\partial^2 z}{\partial x^2} - \dfrac{\partial^2 z}{\partial y^2} = 0\)
is :
Answer: C
The solution is \(z = \phi_{1}(x + y) + \phi_{2}(x - y)\).
This is the wave equation \(z_{xx} = z_{yy}\). Its auxiliary equation \(m^{2} - 1 = 0\) has distinct roots \(m = \pm 1\), so the general solution is d'Alembert's: an arbitrary function of \(x + y\) plus an arbitrary function of \(x - y\).
Check: for \(z = \phi_{1}(x + y)\), \(z_{xx} = \phi_{1}''\) and \(z_{yy} = \phi_{1}''\), so they cancel; likewise for \(\phi_{2}(x - y)\). Factors of \(x\) or \(y\) in front belong to repeated roots and do not satisfy this equation.
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