Use the following results:
$$(l+1)P_{l+1}(x)-x(2l+1)P_l(x)+lP_{l-1}(x)=0,$$ $$P_l(x)+2xP'_l(x)=P'_{l+1}(x)+P'_{l-1}.$$
in order to show the following recurrence relations:
$$(2l+1)P_l(x)=P'_{l+1}(x)-P'_{l-1}(x),$$ $$lP_l(x)=xP'_l(x)-P'_{l-1}(x).$$
Then, with all this relations show that the Legendre polynomials rectify the Legendre equation:
$$\frac{\mathrm{d}}{\mathrm{d}x}\left[\left(1-x^2\right)P'_l(x)\right]+l(l+1)P_l(x)=0.$$
Solution: I've already proved the recurrence relations. The first one is proved taking the first derivative of the first result and the second one use the first relation with the second result. Nevertheless, I couldn't prove that the polynoimals rectify the Legendre equation. I tried substituting $P'_l(x)$ and $lP_l(x)$ on the Legendre eqution using the first and the second relation, but I only complicated the equation. I am grateful for anyhelp!