# Ordinary Differential Equations/Legendre Equation

${\displaystyle {d \over dx}\left[(1-x^{2}){d \over dx}P_{n}(x)\right]+n(n+1)P_{n}(x)=0.}$
The Legendre differential equation may be solved using the standard power series method. The equation has regular singular points at ${\displaystyle x=\pm 1}$ so, in general, a series solution about the origin will only converge for ${\displaystyle \left\vert x\right\vert <1}$. When n is an integer, the solution ${\displaystyle P_{n}\left(x\right)}$ that is regular at ${\displaystyle x=1}$ is also regular at ${\displaystyle x=-1}$, and the series for this solution terminates (i.e. is a polynomial).
The solutions for ${\displaystyle n=0,1,2\dots }$ (with the normalization ${\displaystyle P_{n}\left(1\right)=1}$) form a polynomial sequence of orthogonal polynomials called the Legendre polynomials. Each Legendre polynomial ${\displaystyle P_{n}\left(x\right)}$ is an nth-degree polynomial. It may be expressed using Rodrigues' formula:
${\displaystyle P_{n}(x)={1 \over 2^{n}n!}{d^{n} \over dx^{n}}\left[(x^{2}-1)^{n}\right].}$