Notice that , and .
Substitute these series into the ODE:
Shift the indices to get each series to include a term.
Remove the first term from both series starting at to get all the series starting on the same index :
The coefficients are given recursively by
Let's see if we can find a pattern to these coefficients.
and so on, suggesting that
which is also consistent with . However,
but we can adjust for this easily:
Now all the terms following resemble an exponential series: