View source for Legendre differential equation

The Legendre differential equation is the second order [[ordinary differential equation]] (ODE) which can be written as:
:<math>(1-x^2)d^2y/dx^2-2xdy/dx+l(l+1)y=0\,</math>
which when rearranged to:
:<math>{d \over dx }[(1 - x^2){dy \over dx }]+l(l+1)y=0\,</math> is called Legendre differential equation of order <math>l</math>, where the quantity <math>l</math> is a constant.

:<math>Ly=0\,</math>
where <math>L\,</math> is the Legendre operator:
:<math>L={d \over dx }[(1 - x^2){d \over dx }]+l(l+1)\,</math>

In principle, <math>l</math> can be any number, but it is usually an integer.

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


:<math>y= \sum_{n=0}^{\infty}a_n x^n</math>,
:<math>y' = \sum_{n=0}^{\infty}n a_n x^{n-1}</math>,
:<math>y'' = \sum_{n=0}^{\infty}n(n-1) a_n x^{n-2}</math>.


Substituting these terms into the original equation, one obtains


:{| border="0" cellpadding="0" cellspacing="0"
|-|
|<math>0=Ly\,</math>
|<math> = \big(1-x^2)y'' -2xy'+l(l+1)y</math>
|-
|
|<math>=(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</math>
|-
|
|<math>=\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}</math>
|-
|
|<math>=\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</math>
|-
|
|<math>=\sum_{n=0}^{\infty}\left[(l+n+1)(l-n)a_n + (n+2)(n+1)a_{n+2}\right]x^n</math>.
|-
|}


Thus


:<math> a_2 = -{l(l+1) \over 2} a_0</math>,

and in general,

:<math> a_{n+2} = -{(l+n+1)(l-n) \over (n+2)(n+1)}a_n  </math>.

This series converges when

:<math>\lim_{n \to \infty}\left|{a_{n+2}x^{n+2} \over a_nx^n}\right|<1</math>.


Therefore the series solution has to be cut by choosing:


:<math> n =l \mbox { or } n = -(l+1)\,</math>.


The series cut in specific integers <math>l</math> and <math>l+1</math> produce polynomials called '''Legendre polynomials'''.

==External links==

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[[Category:Differential equations]]
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[[Category:Resources last modified in May 2019]]