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
y = x f ( d y d x ) + g ( d y d x ) . {\displaystyle y=xf\left({\frac {dy}{dx}}\right)+g\left({\frac {dy}{dx}}\right).}
After differentiating once, and rearranging with p = d y / d x {\displaystyle p=dy/dx} , 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 a first order linear differential equation:
d x d p + f ′ ( p ) f ( p ) − p x = − g ′ ( p ) f ( p ) − p {\displaystyle {\frac {dx}{dp}}+{\frac {f'(p)}{f(p)-p}}x={\frac {-g'(p)}{f(p)-p}}}
as the general form:
d x d p + R ( p ) x = Q ( p ) {\displaystyle {\frac {dx}{dp}}+R(p)x=Q(p)}
When f ( p ) = p {\displaystyle f(p)=p} , d'Alembert's equation is reduced to Clairaut's equation.
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