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True or false: a partial differential equation of order one always has a solution containing exactly one arbitrary constant.

  • A.True, provided the equation is linear
  • B.True, unless a singular solution also exists
  • C.False, the complete integral of a first-order equation carrying two
  • D.True, one constant arising from the single integration

Answer: C

The complete integral of a first-order equation in two independent variables carries two arbitrary constants, one for each variable, and the general solution replaces them by an arbitrary function. Counting one constant per order is the ordinary-equation habit, and it does not carry across

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