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The complete integral of \( z = px + qy + pq \) is

  • A.\( z = ax + by + a + b \)
  • B.\( z = a(x + y) + a^{2} \)
  • C.\( z = ax + by + ab \)
  • D.\( z = ax + by \)

Answer: C

The equation is in Clairaut's form, so \( p \) and \( q \) are replaced by constants directly:

\[ z \;=\; ax + by + ab. \]

Substituting back, \( p = a \) and \( q = b \), and the right-hand side reads \( ax + by + ab \) as required. Dropping the product term would satisfy a different equation, \( z = px + qy \)

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