Legendre differential equation

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

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

which can be written as:

${d \over dx }[(1 - x^2){dy \over dx }]+l(l+1)y=0\,$

$Ly=0\,$

where $L\,$ is the Legendre operator:

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

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.

$y= \sum_{n=0}^{\infty}a_n x^n$ ,

$y' = \sum_{n=0}^{\infty}n a_n x^{n-1}$ ,

$y'' = \sum_{n=0}^{\infty}n(n-1) a_n x^{n-2}$ .

Substituting these terms into the original equation, one obtains

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