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:

rearranging 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?

Thanks