n-order perturbative solution of the inhomogeneous wave equation

The exact solution of the inhomogeneous wave equation in one dimension, when the square of the velocity is a linear function of the position, can be written in terms of Bessel functions of the first kind. We use this solution as the zero order approximation for a perturbation expansion and apply it to the case when the square of the velocity can be written as a polynomial in the position. The first and second order perturbation terms, corresponding to quadratic and cubic terms for the square of the velocity, are obtained. A closed formula for the n-order correction in terms of integrals of the Bessel functions of the first kind was also explicitly obtained, this expression can be solved analytically for the first and second order corrections and numerically for higher terms.