I am having trouble solving:
$a\frac{x_{n-1}}{n-1}-\frac{x_{n}}{n}+(1-a)\frac{x_{n+1}}{n+1}=0$
So far I have tried eliminating the n-1,n,n+1 terms by multiplying them out but that doesn't feel right. Intuitively I think the solution would be to set some new variable $y_n=\frac{x_n}{n}$ and do it as usual with the characteristic equation but I'm not sure if I can do that.
Thanks.
edit: Background:I've just started difference equations and need it for a particular probability problem