Clairaut's equation
= Clairaut's equation
{c}
{title2=$y=xy'+f(y')$}
For a twice differentiable solution, put $p=y'$. Differentiation gives $[x+f'(p)]p'=0$. 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 $x=-f'(p)$ and $y=f(p)-pf'(p)$ 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.