We want to find a solution such that
With these conditions, and .
Substituting the series into the ODE gives
so that the coefficients of satisfy
for , or
for . Notice that this implies a dependency of all beyond on , while takes on whatever initial value is given. In particular,
and so on up to
So we can extract two fundamental solutions such that , where
Recall that
which tells us
but is a constant solution and already accounts for the constant term in , and can be reduced to a simpler constant , leaving us with
The Wronskian is
so the two solutions are indeed independent as long as neither initial value is 0.