D'Alembert's equation

In mathematics, d'Alembert's equation is a first order nonlinear ordinary differential equation, named after the French mathematician Jean le Rond d'Alembert. The equation reads as[1]

${\displaystyle y=xf(p)+g(p)}$

where ${\displaystyle p=dy/dx}$. After differentiating once, and rearranging we have

${\displaystyle {\frac {dx}{dp}}+{\frac {xf'(p)+g'(p)}{f(p)-p}}=0}$

The above equation is linear. When ${\displaystyle f(p)=p}$, d'Alembert's equation is reduced to Clairaut's equation.