Help meh pls
2 answers:
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Let 
, so that
Substituting into the ODE gives
The first series starts with a linear term, while the other two start with a constant. Extract the first term from each of the latter two series:
Finally, to get the series to start at the same index, shift the index of the first two series by replacing
with 
. Then the ODE becomes
which can be consolidated to get
![\displaystyle\sum_{k\ge1}\bigg[(3k(k+1)+6(k+1))a_{k+1}+a_k\bigg]x^k+6a_1+a_0=0](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Csum_%7Bk%5Cge1%7D%5Cbigg%5B%283k%28k%2B1%29%2B6%28k%2B1%29%29a_%7Bk%2B1%7D%2Ba_k%5Cbigg%5Dx%5Ek%2B6a_1%2Ba_0%3D0)
![\displaystyle\sum_{k\ge1}\bigg[3(k+1)(k+2)a_{k+1}+a_k\bigg]x^k+6a_1+a_0=0](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Csum_%7Bk%5Cge1%7D%5Cbigg%5B3%28k%2B1%29%28k%2B2%29a_%7Bk%2B1%7D%2Ba_k%5Cbigg%5Dx%5Ek%2B6a_1%2Ba_0%3D0)
You're fixing the solution so that it contains the origin, which means
which in turn means
. With the given recurrence, it follows that 
for all 
, so the solution would be 
. This is to be expected, since 
is clearly a singular point for the ODE.
Substituting into the ODE gives
The first series starts with a linear term, while the other two start with a constant. Extract the first term from each of the latter two series:
Finally, to get the series to start at the same index, shift the index of the first two series by replacing
which can be consolidated to get
You're fixing the solution so that it contains the origin, which means
which in turn means