Given a non-negative integer $\displaystyle n$ we define $\displaystyle P_n(x)$ to be a solution* to:

$\displaystyle (1-x^2)y''-2xy'+n(n+1)y=0 \mbox{ on }(-1,1)$.

Show that:

$\displaystyle \int_{-1}^1P_n(x)P_m(x) dx = 0 \mbox{ for }n\not = m$.

*)Yes, I realize what I said is notwell-definedbut it makes no difference. Pickanysolution.