d’Alembert’s equation

The first differential equationMathworldPlanetmath


is called d’Alembert’s differential equation; here φ and ψ some known differentiableMathworldPlanetmathPlanetmath real functions.

If we denote  dydx:=p, the equation is


We take p as a new variable and derive the equation with respect to p, getting


If the equation  p-φ(p)=0  has the roots  p=p1, p2, …, pk, then we have  dpνdx=0  for all ν’s, and therefore there are the special solutions


for the original equation.  If  φ(p)p, then the derived equation may be written as


which linear differential equation has the solution  x=x(p,C)  with the integration constant C.  Thus we get the general solution of d’Alembert’s equation as a parametric


of the integral curves.

Title d’Alembert’s equation
Canonical name DAlembertsEquation
Date of creation 2013-03-22 14:31:05
Last modified on 2013-03-22 14:31:05
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 16
Author pahio (2872)
Entry type Derivation
Classification msc 34A05
Synonym Lagrange equationMathworldPlanetmath
Related topic ClairautsEquation
Related topic ContraharmonicProportion
Related topic DerivativeAsParameterForSolvingDifferentialEquations