Discussion
A given partial differential equation can have :
(I) no integral surface passing through a given curve
(II) unique integral surface passing through a given curve
(III) infinitely many integral surfaces passing through a given curve
Then :
Answer: D
All three, (I), (II) and (III), are true.
For a first-order PDE and a given initial curve, the Cauchy problem can turn out three ways, depending on how the curve sits relative to the characteristics:
- If the curve is nowhere characteristic (the transversality determinant is nonzero), there is a unique integral surface through it (II).
- If the curve is a characteristic curve, infinitely many integral surfaces pass through it (III).
- If the projection is characteristic but the data along it are not compatible, no integral surface passes through it (I).
For example, for \(z_{x} = 0\) the curve \(x = t,\ y = 0,\ z = t\) admits no solution, the curve \(x = 0,\ y = t,\ z = t\) gives exactly \(z = y\), and the curve \(x = t,\ y = 0,\ z = 0\) lies on every surface \(z = \phi(y)\) with \(\phi(0) = 0\). Options listing only two cases leave one out.
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