Convert the power series solution of $(1+x^2)y''+4xy'+2y=0$ into simple closed-form expression $(a)$Use two power series in $x$ to find the general solution of
$$(1+x^2)y''+4xy'+2y=0$$
and state the set of $x$-values on which each series solution is valid.
$(b)$ Convert the power series solutions in $(a)$ into simple closed-form expressions.
$(c)$ Use $(b)$ to find the general solution of the equation above on the whole real line.
For $(a)$, I used  $y(x)=\sum\limits_{n=0}^\infty a_nx^n$,and solved the recurrence relation to be $a_{2k} = (-1)^ka_0$ and $a_{2k+1} = (-1)^ka_1$, where $a_0$ and $a_1$ are arbitrary. 
And for the solution I got $y(x)=a_0\sum\limits_{k=1}^\infty (-1)^kx^{2k} +a_1\sum\limits_{k=1}^\infty (-1)^kx^{2k+1}$.
How do I convert the solution into simple closed-from expressions, and how should I solve $(c)$?
Thanks for any help.

Hint: Your power series are both geometric series.
