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

I was given a "classical" homework question where I have to prove that the Legendre polynomials solve the differential equation:

$\frac{d}{dx}[(1-x^2)\frac{d}{dx}P_n(x)] + n(n+1)P_n(x) = 0$

However, I was asked to show this using the mathematical induction. I tried to do this using Rodrigues' formula or with the recursion relation:

$(n+1)P_{n+1}(x) = (2n+1)xP_n(x) - nP_{n-1}(x)$ It's probably a quite simple problem, but I can't see the proper way to show that. I'll appreciate any hint!

There's three recurrence relations that help here:

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

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

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

The induction step then looks as follows:


Q.E.D.