I would like to find solution to following ODE: $$\frac{1}{\sin^{n-1}(x)}\frac{d}{dx}(\sin^{n-1}(x)\frac{df}{dx})-\frac{(n-1)f}{\sin^2(x)}=0$$ where $0<x\leq1,\ n\geq 2$ and $f(0)=0$. Is there any general way to solve this kind ODE? what happens if I substitute $\sin(x)$ by general functions. Any hint and reference will be appreciated.
Remark: As many of you have realised, this ODE comes from calculating the eigenfunctions of Laplacian on sphere, but actually, I care more about the following quantity (it is Steklov eigenvalue if you have seen it): $$\frac{f'}{f}\bigg|_{x=1}$$