you might also read the following article:
Method of undetermined coefficients - Wikipedia, the free encyclopedia
in particular the following paragraph:
f a term in the above particular solution for y appears in the homogeneous solution, it is necessary to multiply by a sufficiently large power of x in order to make the two solutions linearly independent. If the function of x is a sum of terms in the above table, the particular solution can be guessed using a sum of the corresponding terms for y.