For some weeks have been studying Legendre polynomial as a solution to this equation. $$ (1-x^2)\frac{d^2}{dx^2}f(x)-2x\frac{d}{dx}f(x)+n(n+1)f(x)=0.$$
I've found them very interesting to learn from purely mathematical perspective but I haven't come across their specific use. Why are they so important and any example suggesting the use of Legendre polynomial would be appreciated.
Thank you.