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 .

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