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Lagrange's linear equation \( Pp + Qq = R \) is solved from the subsidiary equations

  • A.\( P\,dx = Q\,dy = R\,dz \)
  • B.\( \dfrac{dx}{Q} = \dfrac{dy}{P} = \dfrac{dz}{R} \)
  • C.\( \dfrac{dx}{P} = \dfrac{dy}{Q} = \dfrac{dz}{R} \)
  • D.\( \dfrac{dx}{R} = \dfrac{dy}{Q} = \dfrac{dz}{P} \)

Answer: C

The equation says the normal to the surface is perpendicular to the vector \( (P, Q, R) \), so that vector lies in the surface and its direction gives the characteristic curves:

\[ \frac{dx}{P} \;=\; \frac{dy}{Q} \;=\; \frac{dz}{R}. \]

Two independent integrals \( u = a \) and \( v = b \) of these give the general solution \( \phi(u,v) = 0 \). Each denominator must sit under the differential of its own variable

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