# Ordinary Differential Equations/Legendre Equation

${d \over dx} \left[ (1-x^2) {d \over dx} P_n(x) \right] + n(n+1)P_n(x) = 0.$
The Legendre differential equation may be solved using the standard power series method. The equation has regular singular points at $x=\pm1$ so, in general, a series solution about the origin will only converge for $\left\vert x \right\vert < 1$. When n is an integer, the solution $P_n\left(x\right)$ that is regular at $x = 1$ is also regular at $x=-1$, and the series for this solution terminates (i.e. is a polynomial).
The solutions for $n = 0, 1, 2 \dots$ (with the normalization $P_n\left(1\right)=1$) form a polynomial sequence of orthogonal polynomials called the Legendre polynomials. Each Legendre polynomial $P_n\left(x\right)$ is an nth-degree polynomial. It may be expressed using Rodrigues' formula:
$P_n(x) = {1 \over 2^n n!} {d^n \over dx^n } \left[ (x^2 -1)^n \right].$