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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): 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:
${\displaystyle P_{n}(x)={1 \over 2^{n}n!}{d^{n} \over dx^{n}}\left[(x^{2}-1)^{n}\right].}$