If we write this in terms of the operator D we get
From here we see that the complimentary solution is
Now find the particular solution we need to annihilate the right hand side.
Using the Annihilator that you have above gives
This "new" equation have solution
Now we need to remove any terms that are in the complimentary solution. If we relabel the constants we get
From here it is just finding A and B.