Eigenfunction expansion in Legendre polynomials

[hi10]

1. The problem statement, all variables and given/known data

How to use eigenfunction expansion in Legendre polynomials to find the bounded solution of
(1-x^2)f'' - 2xf' + f = 6 - x - 15x^2 on -1<= x <= 1

2. Relevant equations

eigenfunction expansion

3. The attempt at a solution

[r(x)y']' + [ q(x) + λ p(x) ] = f(x)
In this case, r = 1-x^2 , q = 1 , p = 0 , f = 6 - x -15 x^2 , r(-1) = r (1) = 0

Thanks for any help!

[HallsofIvy]

First what are the eigenfunctions you want to use? In other words, what are the solutions to Legendre's equation.

[mulvi]

I have the same problem.

the attachment from hi10 is what i was thinking.

Does anyone know what u(x) is in hi10's post for determining what an is?