How proper is it to turn an ODE into a wave equation?

Question

I have seen the following method used a few times for finding solutions of wave equations.

1. Take an ODE with a known solution, of the form $y''(x) + g(y(x), x) = 0$
2. Switch it to a wave equation of the form $\Box y(x) + g(y(x), x) = 0$, where $x = (t, x, y, z, ...)$
3. Make the substitution in the known solution from $y(x)$ to $y(x.k)$, with $x.k = k_t t - k_x x - k_y y - ...$
4. Make a dispersion relation for the quantity $k^2$ so that the equation is obeyed.

How universally does this method work? What conditions are required to use it? Does it work often enough to simply be a decent heuristic to solve such equations?