To solve Clairaut's equation, one differentiates with respect to , yielding
so
Hence, either
or
In the former case, for some constant . Substituting this into the Clairaut's equation, one obtains the family of straight line functions given by
the so-called general solution of Clairaut's equation.
The latter case,
defines only one solution , the so-called singular solution, whose graph is the envelope of the graphs of the general solutions. The singular solution is usually represented using parametric notation, as , where .
The parametric description of the singular solution has the form
where is a parameter.