To solve such an equation, we differentiate with respect to x, yielding
In the former case, C = dy/dx for some constant C. Substituting this into the Clairaut's equation, we have the family of 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 represents dy/dx.
This equation was named after Alexis Clairaut.
A first-order partial differential equation is also known as Clairaut's equation or Clairaut equation: