You're doing great!
You're probably making simple errors here and there.Find a particular solution of: .
The homogeneous solution is: . . . . . Right!
Now, my first guess for the particular solution is: .
I have all terms repeated here, so multiply by x to all, obtains:
Would that be the right guess? . . . . Yes!
It took me a few tries to get it right . . .
I finally got: .