![\displaystyle\sum_{n\ge0}\bigg[2(n+r)(n+r-1)+3(n+r)-1\bigg]a_nx^{n+r}+\sum_{n\ge0}\bigg[-2(n+r)(n+r-1)-3(n+r)\bigg]a_nx^{n+r-1}=0](/tpl/images/0360/2267/f5295.png)
, we have
(in your case, swap 
and 
before submitting).
by replacing 
, then consolidate the sums to get![\displaystyle\sum_{n\ge1}\bigg[2(n+r-1)(n+r-2)+3(n+r-1)-1-2(n+r)(n+r-1)-3(n+r)\bigg]a_nx^{n+r-1}](/tpl/images/0360/2267/77130.png)
![=\displaystyle x^r\sum_{n\ge1}\bigg[-4n-4r\bigg]a_nx^{n-1}](/tpl/images/0360/2267/9bfcc.png)
, we then have this as
are related.
Need with indicial equation and odes
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