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

${\displaystyle y=xf\left({\frac {dy}{dx}}\right)+g\left({\frac {dy}{dx}}\right).}$

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

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

The above equation is a first order linear differential equation:

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

as the general form:

${\displaystyle {\frac {dx}{dp}}+R(p)x=Q(p)}$

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

References

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