Clairaut's equation

In mathematics, a Clairaut's equation is a differential equation of the form

___(1)

where we define

Differentiating (1) with respect to x, we have

in other words,

as

is non zero due to x being non zero.

From the former equation, p = C, for some constant C, and substituting this into Clairaut's equation, we have

This is the general solution of Clairaut's equation. The special feature of such equations is that there are also singular solutions, which are envelopes of graphs of general solutions.