Solving (x²)y″+y=0 with power series

[momen salah]

hi guys it's me you helped me last time in my bonus problem(thank's for that) i need help again naw please it's a two marks bonus problem:

solve using power series:
(x^2)(y")+y=0

after solving it i stopped at :

an[n^2-n+1]=0

 

[HallsofIvy]

Try a solution of the form
$$y= \sum_{n=0}^{\infty} a_nx^{n+i}$$

where i is not necessarily positive nor even an integer. Look at the lowest powers after you have put that into the differential equation (and assume a₀ is NOT 0) in order to determine i.