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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

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

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.

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.

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