Series solution about a singular point--help with y(2)

The given equation is x(x-1)y'' + 3y' - 2y = 0.

I was able to discern that a(k+1) = a(k) * [(k+r)(k+r-1)-2]/[(k+r+1)(k+r-3)], and the roots are 0 and 4. That recursive equation gives me the correct coefficients for a(0) through a(3), and of course, a(4) is undefined. But I can't get it right from a(5) on. I used the root 0 to get those first few a's; was I supposed to switch over to 4?

Re: Series solution about a singular point--help with y(2)

Regard the $\displaystyle a_{4}$ coefficient from the $\displaystyle r=0$ case as being arbitrary and then calculate the remaining coefficients in terms of $\displaystyle a_{4}.$

That gets you a solution containing two arbitrary constants, $\displaystyle a_{0}$ and $\displaystyle a_{4},$ in which case you have the general solution of the differential equation.

If you generate the series based on $\displaystyle r=4,$ you will find that it duplicates the second part of the $\displaystyle r=0$ solution

Re: Series solution about a singular point--help with y(2)

Oh goodness--I made the remaining terms in terms of a(5), not a(4). Changing that fixed the problem.