$(4$ points$)$ The power series solution to the Initial Value Problem $($IVP$)$

$(x^2-4)y^{\text{'}\text{'}}8x{y}^{\text{'}}6y=0,y(0)=1,y^{\text{'}}(0)=0$

is given by

$y(x)={\displaystyle \underset{n=0}{\overset{}{}}}{a}_n{x}^{2n}{\displaystyle \underset{n=0}{\overset{}{}}}{b}_n{x}^{2n1}$

where

$a_n=$

$b_n=$