Find the first six non-zero terms of the power series solution at

x = 0 of the following IVP.

$\displaystyle (x^2 + 1)y'' + xy' + 2xy = 0; y(0) = 2; y'(0) = 3$

So far I have:

supposing that $\displaystyle y=C_0+C_1x+C_2x^2+....$is the power series solution I have solved for $\displaystyle C_0=2$.

and from $\displaystyle y'=C_1+2C_2x+3C_3x^2+....$I have solved for $\displaystyle C_1=3$

Using $\displaystyle y= \displaystyle\sum_{n=0}^{\infty}\ {C_nX^n}$

I have got first and second derivative of the sum and have plugged into the original equation and got:

$\displaystyle y= \displaystyle\sum_{n=2}^{\infty}\ {n(n-1)C_nX^n}+ \displaystyle\sum_{n=2}^{\infty}\ {n(n-1)C_nX^(n-2)}+ \displaystyle\sum_{n=1}^{\infty}\ {nC_nX^n} - 2\displaystyle\sum_{n=0}^{\infty}\ {C_nX^(n+1)}=0$

But now I get confused as to what to do next to have the same power of x as what my sum begins with.

Can anyone tell me how I could break down my sums to solve for my 6 terms.

Thank you!!!