Here is the Legendre differential equation:
$$ (1-x^2)\frac{d^2y}{dx^2}-2x\frac{dy}{dx}+l(l+1)y=0$$
It's well known that it can be solved using power series method but I wonder if there is another approach for solving that. I searched on the Internet and only found following link: How to solve Legendre's differential equation without power series assumption?
but that answer isn't really satisfactory. If we let $l = 1$, finding two linearly independent isn't difficult: Find two linearly independent solutions of a Legendre equation about $x=0.$
