D'Alembert's Differential Equation

D'Alembert's (or Lagrange’s) differential equation has the form

, (1)

where . Differentiating the equation with respect to , we get

. (2)

This equation is linear with respect to :
. (2')

From this, we get the solution

. (3)

Here is a solution of the corresponding homogeneous equation of (2’) and is a particular solution of (2’). Equations (1) and (3) determine the solution parametrically. Eliminating the parameter (if possible), we get the general solution in the form .