given that the problem requires a lot of detail. i will only be able to give you some hints of what to do
Power Series: y(x) = ∑n=0 anxn
This approach requires you to assume that y(x) has a power series form. Now you need to find the relevant derivatives y'(x) and y''(x)
Note on derivatives
y'(x) = ∑n=? n anxn-1
Note that if the index were zero the first term would be 1/x. So that would be a problem. Therefore n=1 is the first index. A similar thing happens for the second derivative
y'(x) = ∑n=1 n anxn-1
y''(x) = ∑n=2 n (n-1) anxn-2
Now the goal is to replace this derivatives on the differential equation and find conditions on the coefficients. Note that there will be 2 coefficients that you need to provide as boundary conditions since is a second order differential equation.
Hope this helps!