Discussion
The partial differential equation \(yp - xq = 0\), has :
Answer: A
There is a unique integral surface through \(x^{2} = z,\ y = 0\).
The general solution of \(yp - xq = 0\) is \(z = \phi(x^{2} + y^{2})\), because the characteristics satisfy \(dx/y = dy/(-x)\), giving \(x^{2} + y^{2} = \text{const}\), and \(z\) is constant along them.
On the curve, \(y = 0\) and \(z = x^{2}\), so \(\phi(x^{2}) = x^{2}\), that is \(\phi(t) = t\) for every \(t \ge 0\). This fixes \(\phi\) on all the values \(x^{2} + y^{2}\) can take, so
\[ z = x^{2} + y^{2} \]
is the only solution, a paraboloid. The curve meets each characteristic circle exactly where it crosses the \(xz\)-plane, which is why the data determine \(\phi\) once and for all. So 'finitely many', 'infinitely many' and 'none' are wrong.
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