Discussion
Consider the following two statements :
(I) Every surface generated by a one parameter family of characteristics is an integral surface
(II) Every integral surface is generated by the characteristic curves
Then :
Answer: C
(I) and (II) are both true.
For the quasi-linear equation \(Pp + Qq = R\), the characteristic curves are the integral curves of the direction field \((P, Q, R)\).
(I) A surface swept out by a one-parameter family of characteristics has \((P, Q, R)\) tangent to it at every point, which is exactly the condition \(Pp + Qq = R\). So it is an integral surface.
(II) Conversely, through every point of an integral surface the characteristic curve through that point lies on the surface (a standard theorem in Sneddon's treatment), so every integral surface is a union of characteristics.
These two facts together are the basis of the method of characteristics, so the options denying either one are wrong.
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