Discussion
Which one of the following is not the general solution of the partial differential equation :
\[ z(xp - yq) = y^{2} - x^{2} \]
Answer: C
The relation that is not a general solution is x2 + y2 + z2 = f(y/x).
Lagrange's auxiliary equations are \(\frac{dx}{xz} = \frac{dy}{-yz} = \frac{dz}{y^{2} - x^{2}}\). The first pair gives \(\frac{dx}{x} + \frac{dy}{y} = 0\), so \(xy = c_{1}\). Using multipliers x, y, z, the numerator \(x \cdot xz - y \cdot yz + z(y^{2} - x^{2}) = 0\), so \(x^{2} + y^{2} + z^{2} = c_{2}\).
The general solution is any relation between \(c_{1} = xy\) and \(c_{2} = x^{2} + y^{2} + z^{2}\). Option A is one directly. Since \((x + y)^{2} + z^{2} = c_{2} + 2c_{1}\), options B and D are also relations between \(c_{1}\) and \(c_{2}\). But \(y/x\) is not constant along the characteristics, so option C is not a general solution.
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