Discussion
The complete integral of the partial differential equation
\(f(p, q) = 0\)
is :
Answer: C
The complete integral is \(z = \phi(a)x + ay + b\).
For an equation in \(p = z_{x}\) and \(q = z_{y}\) alone, try \(z = \alpha x + \beta y + b\). Then \(p = \alpha\) and \(q = \beta\), so the equation holds whenever \(f(\alpha, \beta) = 0\). Taking \(q = a\) and solving for \(p = \phi(a)\) gives a two-parameter family \(z = \phi(a)x + ay + b\), which is the complete integral.
Option A fixes \(q = 1\), leaving only one effective parameter. Options with \(x^{2}\) give \(p = 2\phi(a)x\) or \(p = 2x\), which depend on \(x\) and do not satisfy a relation between constants \(p\) and \(q\).
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