D'Alembert's equation explained

In mathematics, d'Alembert's equation, sometimes also known as Lagrange's equation,^[1] is a first order nonlinear ordinary differential equation, named after the French mathematician Jean le Rond d'Alembert. The equation reads as^[2]

y = x f(p) + g(p)

where

p = dy/dx . After differentiating once, and rearranging we have(dx)/(dp) + (x f'(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.

Notes and References

1. Web site: Weisstein . Eric W. . d'Alembert's Equation . 2024-06-02 . mathworld.wolfram.com . en.
2. Davis, Harold Thayer. Introduction to nonlinear differential and integral equations. Courier Corporation, 1962.