D'Alembert equation

A differential equation of the form

$$y=x\phi(y')+f(y'),$$

where $\phi$ and $f$ are the functions to be differentiated; first studied in 1748 by J. d'Alembert. Also known as the Lagrange equation.

For $\phi(y')=y'$ the d'Alembert equation specializes to the Clairaut equation. For some results on (solving) the d'Alembert equation cf., e.g., [a1].

References

[a1]	E.L. Ince, "Integration of ordinary differential equations" , Oliver & Boyd (1963) pp. 43 {{ZBL}0612.34002}}