Legendre Polynomial and Legendre Equation

[Fantini]

Given $f(x) = (x^2-1)^l$ we know it satisfies the ordinary differential equation $$(x^2-1)f'(x) -2lx f(x) = 0.$$ The book defines the Legendre polynomial $P_l(x)$ on $\mathbb{R}$ by Rodrigues's formula $$P_l(x) = \frac{1}{2^l l!} \left( \frac{d}{dx} \right)^l (x^2-1)^l.$$ I'm asked to prove by $(l+1)$-fold differentiation of the first ODE that $P_l(x)$ satisfies Legendre's equation: $$(1-x^2) u''(x) -2x u'(x) +l(l+1) u(x)=0,$$ for a twice differentiable function $u: \mathbb{R} \to \mathbb{R}$.

Honestly I'm out of ideas. I have differentiated the ODE and obtained another relation, but I didn't see a way to use it. I don't know how to translate $(l+1)$-fold differentiation into something usable, nor know how to actually perform the $(l+1)$-fold differentiation if one were to brute-force it. Please guide me in the right direction. :)

 

[GJA]

Hey Fantini,

This type of nail is best struck with the Leibniz General Product Rule Formula hammer:

Differentiation rules - Wikipedia, the free encyclopedia

You'll want to use this formula to compute

\(\displaystyle \left(\frac{d}{dx} \right)^{l+1}\left[(x^{2}-1)f'(x)-2lxf(x) \right],\)

which we know is zero from the ODE. I can supply the details of this computation if you get stuck along they way, but I think you'll see how to proceed on your own. Let me know if anything is unclear/not quite right.