Ok here's a funny ODE to solve:
clearly a straight forward power series substitution won't work here since we have a regular singularity at x = 0
so try the frobenius method by expanding around x = 0.
Assume is a solution where is some constant.
So we have and
Put this back in:
after some algebra and stuff:
clearly lowest term is with it's coefficient as hence
Now , so
Now we find the coefficients of the term where s is some constant, this gives:
for s = 1, 2, etc
Thus we found a recurrence relationship with and as arbitrary initial values.
A bit of playing around quickly shows that:
Thus we have one of the solutions to be
However because a_0 and a_1 are arbitrary, let us pick.... , now magically we have:
So is one of the basis for the general solution of this ode.
Now I was wondering, since and are arbitrary, then would ANY ( ) and work? Say and which then implies that there is an "infinite" number of different basis for the general solution of this ode?