The characteristics of the PDE are x \pm ct = \mathrm{const} (where \pm sign states the two solutions to quadratic equation), so we can use the change of variables \mu = x + ct (for the positive solution) and \eta = x-ct (for the negative solution) to transform the PDE to u_{\mu\eta} = 0. The general solution of this PDE is u(\mu,\eta) = F(\mu) + G(\eta) where F and G are C^1 functions. Back in x, t coordinates,
