In mathematics, a Clairaut's equation is a differential equation of the form Mathematics is the study of quantity, structure, space and change. ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ...

To solve such an equation, we differentiate with respect to x, yielding

so

Hence, either

or

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. A singular solution of a differential equation is a solution that satisfies the following conditions: It solves the original differential equation. ... An envelope is a flat, folded paper container often used for mailing things. ...