Clairaut's equation ($y=x{y}^{\prime }+f(y^{\prime })$)

For a twice differentiable solution, put $p=y^{\prime }$. Differentiation gives $[x+f^{\prime }(p)]p^{\prime }=0$. One family consists of straight lines $y=Cx+f(C)$ with constant slope $C$. The other consists of the envelope of those lines, given parametrically by $x=-f^{\prime }(p)$ and $y=f(p)-p{f}^{\prime }(p)$ where these equations define a differentiable graph with derivative $p$. An envelope is a singular solution not obtained by fixing $C$. If only continuous first derivatives are required, a solution may also join a line to its envelope at their point of tangency, provided the joined function satisfies the original equation. Such joins need not have a continuous second derivative.