D'Alembert's equation

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

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

where p = d y / d x {\displaystyle p=dy/dx} . After differentiating once, and rearranging we have

d x d p + x f ′ ( p ) + g ′ ( p ) f ( p ) − p = 0 {\displaystyle {\frac {dx}{dp}}+{\frac {xf'(p)+g'(p)}{f(p)-p}}=0}

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

References

  1. ^ Davis, Harold Thayer. Introduction to nonlinear differential and integral equations. Courier Corporation, 1962.


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