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

$y=xf(p)+g(p)$

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

$\frac{dx}{dp}+\frac{x{f}^{\prime }(p)+g^{\prime }(p)}{f(p)-p}=0$

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