I'm taking an ODE course at the moment, and my instructor gave us the following problem:
Derive the following formula for Legendre functions $Q_n(x)$ of the second kind:
$Q_n(x) = P_n(x) \int \frac{1}{[P_n(x)]^2 (1-x^2)}dx$
where $P_n(x)$ is the $n$-th Legendre polynomial.
He introduced Legendre functions in the context of second order ODEs, but we haven't really used them for anything - moreover, this is the only problem we were assigned that has anything to do with them. As a result, I'm sort of at a loss of where to start.
I've tried a couple of things (like using the actual Legendre ODE
$(1-x^2)y^{\prime \prime} - 2xy^{\prime} + n(n+1)y = 0$
and plugging in the solution $y(x)=a_1P_n(x)+a_2Q_n(x)$ and proceeding from there) but so far, haven't been able to go anywhere.
Any help (preferably as elementary as possible) would be much appreciated. Thanks!