I am currently confronted with the following equation $$
0=w''(t)(t^2-t)+w'(t)((2n-1)t^2-n)+w(t)(n-1)^2t
$$ for $t\in(-1,1)$. There following assumption is also in force, $w(t)\in C^{2}(-1,1)$

I would like to prove uniqueness of the above solution (so $w=0$). 
I have found some papers which prove uniqueness of such a class of equations with coefficients that contain singularities but none that are applicable in my case. Does anyone have an idea?