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 $t\in(-1,1)$. So $w:(-1,1)\rightarrow\mathbb{R}$. The following assumption is also in force, $w(t)\in C^{2}(-1,1)$

I would like to prove that the solution to the above problem is unique (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?

Edit: I apologise for not having added the following. n is an integer greater than or equal to 2.