mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, d'Alembert's equation is a first order nonlinear

ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...

, named after the French mathematician

Jean le Rond d'Alembert Jean-Baptiste le Rond d'Alembert (; ; 16 November 1717 – 29 October 1783) was a French mathematician, mechanician, physicist, philosopher, and music theorist. Until 1759 he was, together with Denis Diderot, a co-editor of the ''Encyclopédie ...

. The equation reads asDavis, Harold Thayer. Introduction to nonlinear differential and integral equations. Courier Corporation, 1962. :

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

where

p=dy/dx

. After differentiating once, and rearranging we have :

\frac + \frac=0

The above equation is linear. When

f(p)=p

, d'Alembert's equation is reduced to

Clairaut's equation In mathematical analysis, Clairaut's equation (or the Clairaut equation) is a differential equation of the form :y(x)=x\frac+f\left(\frac\right) where ''f'' is continuously differentiable. It is a particular case of the Lagrange differential eq ...

References

Equations of physics Mathematical physics Differential equations Ordinary differential equations {{Mathanalysis-stub