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Clairaut's form of a first-order partial differential equation is

  • A.\( f(p,q) = 0 \)
  • B.\( z = px + qy \) only
  • C.\( f(x,p) = g(y,q) \)
  • D.\( z = px + qy + f(p,q) \)

Answer: D

Clairaut's form is \( z = px + qy + f(p,q) \), whose complete integral is obtained simply by replacing \( p \) and \( q \) by constants \( a \) and \( b \):

\[ z \;=\; ax + by + f(a,b). \]

The bare \( f(p,q) = 0 \) and the separable \( f(x,p) = g(y,q) \) are two of the other standard types, each with its own recipe

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