The equation that we want to solve is

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.