Solving Laguerre DEby translating it into an Euler equation

[Ratpigeon]

Homework Statement

Find the indicial equation and all power series solutions around 0 of the form
x^r Ʃa_n xⁿ for:
x y'' -(4+x)y'+2y=0
- apparently one of these solutions is a laguerre pilynomial

Homework Equations

the indicial equation is the roots of
r(r-1) +p₀r+q₀
where p₀=lim(x->0)( x(-4-x)/x)=-4
and q₀=lim(x->0)( x^2 *2/x)=0
Hence the indicial equation is:
r^2-r - 4r =r(r-5)

The Attempt at a Solution

I have a solution for the root at r=5, but I'm not sure how to do it for r=0, which is the Laguerre one...?

 

[clamtrox]

What is the difficulty here? Just plug your ansatz $y = \sum_n a_n x^n$ into the equation, and solve like usual. By googling "laguerre polynomial" you will see that you expect to get a solution which contains only 2+1 = 3 terms; you will probably find something like $a_{n+1} \propto (n-2) a_n$, meaning that for all n>2, a_n = 0.