Prove that the Legendre polynomial recurrence relationship satisfies the defining differential equation

Question

I am trying to show that from this recurrent relationship $$ (n+1)P_{n+1}(x) = (2n+1)xP_n(x) - nP_{n-1}(x) $$ that the Legendre polynomial $P_n(x)$ satisfies the differential equation $$ (1-x^2)P'' - 2xP' + n(n+1)P = 0 $$

I can show both equations from the corresponding generating function but am not quite sure how to show from the recurrent relationship that the differential equation holds.

Thank you.

Answer 1

See Use induction to prove that Legendre polynomials solve the corresponding differential equation

Although, as mentioned there, I used 3 recurrence relationships, not one (copied below).

$$P_{n+1}^{'} -P_{n-1}^{'} = (2n+1)P_n$$

$$(n+1)P_{n+1} = (2n+1)xP_n -nP_{n-1}$$

$$P_{n+1}-P_{n-1} = (x^2-1)\cdot \frac{2n+1}{n(n+1)}\cdot P_n^{'}$$

I guess that one of these three can be derived from the other two, but I think not that the top and bottom one can both be derived from the middle one - but feel free to give it a try :-).