D'Alembert's (or Lagrange’s) differential equation has the form
where . Differentiating the equation with respect to , we get
This equation is linear with respect to : . (2')
From this, we get the solution
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 .