We first look for a particular solution

We first look for a particular solution for

with .
To find a particular solution for (19), we expand the polynomial in terms of the polynomial basis . That is,
For the equation (19), we consider three cases:
Case 1) If , we look for a particular solution that is in the form of

Case 2) If and , we look for a particular solution that is in the form of

Case 3) If , the particular solution is

We consider case 1) with the assumption in the following since other cases can be handled similarly. We The particular solution (20) is substituted into (19),

By comparing coefficients, the should satisfy the following system of equations