Legendre differential equation

The Legendre differential equation is the second order ordinary differential equation (ODE) which can be written as:

$(1-x^{2})d^{2}y/dx^{2}-2xdy/dx+l(l+1)y=0\,$

which when rearranged to:

${d \over dx}[(1-x^{2}){dy \over dx}]+l(l+1)y=0\,$ is called Legendre differential equation of order $l$, where the quantity $l$ is a constant.

$Ly=0\,$

where $L\,$ is the Legendre operator:

$L={d \over dx}[(1-x^{2}){d \over dx}]+l(l+1)\,$

In principle, $l$ can be any number, but it is usually an integer.

We use the Frobenius method to solve the equation in the region $|x|\leq 1$. We start by setting the parameter p in Frobenius method zero.

$y=\sum _{n=0}^{\infty }a_{n}x^{n}$,

$y'=\sum _{n=0}^{\infty }na_{n}x^{n-1}$,

$y''=\sum _{n=0}^{\infty }n(n-1)a_{n}x^{n-2}$.

Substituting these terms into the original equation, one obtains

$0=Ly\,$	$={\big (}1-x^{2})y''-2xy'+l(l+1)y$
	$=(1-x^{2})\sum _{n=0}^{\infty }n(n-1)a_{n}x^{n-2}-2x\sum _{n=0}^{\infty }na_{n}x^{n-1}+l(l+1)\sum _{n=0}^{\infty }a_{n}x^{n}$
	$=\sum _{n=0}^{\infty }\left[-n(n-1)-2n+l(l+1)\right]a_{n}x^{n}+\sum _{n=0}^{\infty }n(n-1)a_{n}x^{n-2}$
	$=\sum _{n=0}^{\infty }\left[l^{2}-n^{2}+l-n\right]a_{n}x^{n}+\sum _{n=-2}^{\infty }(n+2)(n+1)a_{n+2}x^{n}$
	$=\sum _{n=0}^{\infty }\left[(l+n+1)(l-n)a_{n}+(n+2)(n+1)a_{n+2}\right]x^{n}$.

Thus

$a_{2}=-{l(l+1) \over 2}a_{0}$,

and in general,

$a_{n+2}=-{(l+n+1)(l-n) \over (n+2)(n+1)}a_{n}$.

This series converges when

$\lim _{n\to \infty }\left|{a_{n+2}x^{n+2} \over a_{n}x^{n}}\right|<1$.

Therefore the series solution has to be cut by choosing:

$n=l{\mbox{ or }}n=-(l+1)\,$.

The series cut in specific integers $l$ and $l+1$ produce polynomials called Legendre polynomials.