# Legendre's equation

1. Dec 26, 2008

### KateyLou

1. The problem statement, all variables and given/known data

Obtain the recurrene relation between the coefficient ar in the series solution

y= (between r=0 and $$\infty$$) $$\Sigma$$ arxr

to (1-x2)y''-2xy'+k(k+1)y=0
Deduce that if k is a positive integer, then ak+2=0, so that the equation possesses a solution which is a polynomial of degree k.

2. Relevant equations

(there is more to this question....but i think ill try getting through this bit first!)

3. The attempt at a solution

I have managed to get the correct recurrance relation

(n+2)(n+1)an+2-an(n(n+1)-k(k+1))=0

But i have no idea what to do now (the show if k is a positive integer) etc :-(

2. Dec 26, 2008

### HallsofIvy

So
$$a_{n+2}= a_n\frac{n(n+1)- k(k+1)}{(n+1)(n+2)}$$
Assume that a0 and a1[/sup] are given. What is a2, a3, a4 in terms of a0 and a1. Can you guess a form for the general an form? Can you then prove it by, say, induction?

3. Dec 26, 2008

### KateyLou

i am not too sure what induction is.....(sorry!) guessing a form for an - do you is this not given by the recurrance relation?