Discussion
The general solution of the partial differential equation :
\[ \begin{aligned} &xzp - yzq = y^{2} - x^{2}, \\ &\left(\frac{\partial z}{\partial x} = p, \frac{\partial z}{\partial y} = q\right) \end{aligned} \]
is :
Answer: A
The general solution is \(\phi(xy, x^{2} + y^{2} + z^{2}) = 0\).
This is Lagrange's linear equation \(Pp + Qq = R\) with \(P = xz\), \(Q = -yz\), \(R = y^{2} - x^{2}\). The auxiliary equations are \(\dfrac{dx}{xz} = \dfrac{dy}{-yz} = \dfrac{dz}{y^{2} - x^{2}}\).
From the first two, \(dx/x = -dy/y\), so \(xy = c_{1}\). Using multipliers \(x, y, z\): \(xP + yQ + zR = x^{2}z - y^{2}z + z(y^{2} - x^{2}) = 0\), so \(x\,dx + y\,dy + z\,dz = 0\) and \(x^{2} + y^{2} + z^{2} = c_{2}\).
The general solution is \(\phi(c_{1}, c_{2}) = 0\). The other options pair \(x^{2} + y^{2} + z^{2}\) or \(xyz\) with expressions that are not constant along the characteristics.
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