To solve Clairaut's equation, one differentiates with respect to x, yielding
In the former case, C = dy/dx for some constant C. 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 y(x), 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 (x(p), y(p)), where p = dy/dx.
The parametric description of the singular solution has the form
where t is a parameter
The following curves represent the solutions to two Clairaut's equations:
In each case, the general solutions are depicted in black while the singular solution is in violet.
By extension, a first-order partial differential equation of the form
is also known as Clairaut's equation.