I don't know what method is referred to in "section 4.3", but I'll suppose it's reduction of order and use that to find the exact solution. Take
, so that
and we're left with the ODE linear in
:
Now suppose
has a power series expansion
Then the ODE can be written as
All the coefficients of the series vanish, and setting
in the power series forms for
and
tell us that
and
, so we get the recurrence
We can solve explicitly for
quite easily:
and so on. Continuing in this way we end up with
so that the solution to the ODE is
We also require the solution to satisfy
, which we can do easily by adding and subtracting a constant as needed: