Recall that
Differentiating the power series series for y(x) gives the series for y'(x) :
Now, replace everything in the DE with the corresponding power series:
The series on the right side has no even-degree terms, so if we split up the even- and odd-indexed terms on the left side, the even-indexed
series should vanish and only the odd-indexed
terms would remain.
Split up both series on the left into even- and odd-indexed series:
Next, we want to condense the even and odd series:
• Even:
• Odd:
Notice that the right side of the DE is odd, so there is no 0-degree term, i.e. no constant term, so it follows that
.
The even series vanishes, so that
for all integers k ≥ 1. But since
, we find
and so on, which means the odd-indexed coefficients all vanish,
.
This leaves us with the odd series,
We have
So long as you're given an initial condition
(which corresponds to
), you will have a non-zero series solution. Let
with
. Then
and so the first four terms of series solution to the DE would be