Consider the following differential equation

$2y^{\text{'}\text{'}}+(x+1)y^{\text{'}}+y=0,x_0=2.$

$($a$)$ Seek a power series solution for the given differential equation about the given point $x_0$; find the recurrence relation that the coefficients must satisfy.

$a_{n+2}=\frac{-3}{2(n+2)},a_{n+1}+,a_n,n=0,1,2,$cdots

$($b$)$ Find the first four nonzero terms in each of two solutions $y_1$ and $y_2.$ NOTE: For $y_1,$ set $a_0=1$ and $a_1=0$ in the power series to find the first four non$-$zero terms. For $y_2,$ set $a_0=0$ and $a_1=1$ in the power series to find the first four non$-$zero terms.

X $y_1(x)=$

$y_2(x)=$

Please help solve part a & b