Clairaut's Differential Equation

$$y=x{dy\over dx}+f\left({dy\over dx}\right)$$

or

$$y=px+f(p),$$

where $f$ is a Function of one variable and $p\equiv dy/dx$. The general solution is $y=cx+f(c)$. The singular solution Envelopes are $x=-f'(c)$ and $y=f(c)-cf'(c)$.

See also d'Alembert's Equation

References

Boyer, C. B. A History of Mathematics. New York: Wiley, p. 494, 1968.