Assume a solution of the form

with derivatives


Substituting into the ODE, which appears to be

gives


![(a_0-8a_2)+(4a_1-24a_3)x+\displaystyle\sum_{n\ge2}\bigg[(n+1)^2a_n-4(n+2)(n+1)a_{n+2}\bigg]x^n=0](https://tex.z-dn.net/?f=%28a_0-8a_2%29%2B%284a_1-24a_3%29x%2B%5Cdisplaystyle%5Csum_%7Bn%5Cge2%7D%5Cbigg%5B%28n%2B1%29%5E2a_n-4%28n%2B2%29%28n%2B1%29a_%7Bn%2B2%7D%5Cbigg%5Dx%5En%3D0)
which gives the recurrence for the coefficients
,

There's dependency between coefficients that are 2 indices apart, so we consider 2 cases.
- If
, where
is an integer, then




and so on, with the general pattern

- If
, then




and so on, with

Then the two independent solutions to the ODE are

and

By the ratio test, both series converge for
, which also can be deduced from the fact that
are singular points for this ODE.
1) x=1
2) x=2.9
3) x=-10
4) x=?
5) IDK