for (x^2+2)y'' +3x-y = 0 about the ordinary point x = 0

my guess was y = sum c_nx^n

I think that I did my sum correctly but if someone can double check it for me, that would be great.

<a href="http://www.codecogs.com/eqnedit.php?latex=\sum_{n=2}^{\infty}n(n-1)c_nx^n @plus; 2 \sum_{ n=0}^{ \infty}(n@plus;2)(n@plus;1)c_{n@plus;2}x^{n}@plus; 3 \sum_{ n=1}^{\ \infty} nc_nx^n-\sum_{ n=0}^{ \infty} c_nx^n=0" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\sum_{n=2}^{\infty}n(n-1)c_nx^n + 2 \sum_{ n=0}^{ \infty}(n+2)(n+1)c_{n+2}x^{n}+3 \sum_{ n=1}^{\ \infty} nc_nx^n-\sum_{ n=0}^{ \infty} c_nx^n=0" title="\sum_{n=2}^{\infty}n(n-1)c_nx^n + 2 \sum_{ n=0}^{ \infty}(n+2)(n+1)c_{n+2}x^{n}+3 \sum_{ n=1}^{\ \infty} nc_nx^n-\sum_{ n=0}^{ \infty} c_nx^n=0" /></a>