Just FYI your questions are very vague.

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

Then

and

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.