Hey I need to solve this problem by means of inverse operators:
I think this is what you mean...
Solving for y gives
Since is a polynomial we can express each operator as a geometric series, but 1st partial fractions gives
Combing the sums gives
Note that since the right hand side is only a degree 1 polynomial we only need two terms from the series
So this gives
You still need to use the eigenfunctions to find the complimentary solution.