d'Alembert's equation

Not to be confused with d'Alembert's principle, d'Alembert's formula or the d'Alembert operator.

In mathematics, d'Alembert's equation, sometimes also known as Lagrange'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

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

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