D'Alembert's equation

Template:Short description Template:Distinguish Template:One source 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

$y=xf\left(\frac{dy}{dx}\right)+g\left(\frac{dy}{dx}\right).$

After differentiating once, and rearranging with $p=dy/dx$, we have

$\frac{dx}{dp}+\frac{x{f}^{\prime }(p)+g^{\prime }(p)}{f(p)-p}=0$

The above equation is a first order linear differential equation:

$\frac{dx}{dp}+\frac{f^{\prime }(p)}{f(p)-p}x=\frac{-g^{\prime }(p)}{f(p)-p}$

as the general form:

$\frac{dx}{dp}+R(p)x=Q(p)$

When $f(p)=p$, d'Alembert's equation is reduced to Clairaut's equation.

References

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