tutorial [also look at post #6 and #8 for two other techniques].
Going along with this, we solve the homogeneous solution first:
You should end up with
Now for the particular solution, solve the non homogeneous equation.
This will lead us to finding the annihilator of , which is
The DE becomes with multiplicity six. Thus, the particular solution with only consist of r repeating 4 times, since 2 were used in the homogeneous solution.
Now substitute into the original DE and find the coefficients...
If you still do the annihilator approach,
Try to do something similar to what I have done above.
I hope this helps.