Legendre differential equation

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

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

which when rearranged to:

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

$Ly=0$

where $L$ is the Legendre operator:

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

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 $|x|\le 1$. We start by setting the parameter p in Frobenius method zero.

$y=\sum _{n=0}^{\mathrm{\infty }}{a}_n{x}^n$,

$y^{\prime }=\sum _{n=0}^{\mathrm{\infty }}n{a}_n{x}^{n-1}$,

$y^{″}=\sum _{n=0}^{\mathrm{\infty }}n(n-1)a_n{x}^{n-2}$.

Substituting these terms into the original equation, one obtains

$0=Ly$	$=(1-x^2)y^{″}-2x{y}^{\prime }+l(l+1)y$
	$=(1-x^2)\sum _{n=0}^{\mathrm{\infty }}n(n-1)a_n{x}^{n-2}-2x\sum _{n=0}^{\mathrm{\infty }}n{a}_n{x}^{n-1}+l(l+1)\sum _{n=0}^{\mathrm{\infty }}{a}_n{x}^n$
	$=\sum _{n=0}^{\mathrm{\infty }}[-n(n-1)-2n+l(l+1)]a_n{x}^n+\sum _{n=0}^{\mathrm{\infty }}n(n-1)a_n{x}^{n-2}$
	$=\sum _{n=0}^{\mathrm{\infty }}[l^2-n^2+l-n]a_n{x}^n+\sum _{n=-2}^{\mathrm{\infty }}(n+2)(n+1)a_{n+2}{x}^n$
	$=\sum _{n=0}^{\mathrm{\infty }}[(l+n+1)(l-n)a_n+(n+2)(n+1)a_{n+2}]x^n$.

Thus

$a_2=-\frac{l(l+1)}{2}{a}_0$,

and in general,

$a_{n+2}=-\frac{(l+n+1)(l-n)}{(n+2)(n+1)}{a}_n$.

This series converges when

$\underset{n\to \mathrm{\infty }}{\lim }\left|\frac{a_{n+2}{x}^{n+2}}{a_n{x}^n}\right|<1$.

Therefore the series solution has to be cut by choosing:

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

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