When does a power series ansatz give one or two solutions to an ODE?

[Jerbearrrrrr]

Hi,

could someone please link me to the relevant theorems etc (or explain personally) that answer the issue that follows.

Say you have an ODE (let's say 2nd order for now).
Let's look for a power series solution (ie assume we're engineers).
So, we write out a couple of sigmas etc and sub stuff into the ODE...

If we're lucky, it may turn out that, say, all the coefficients of odd powers depend on an arbitrary constant, and all the coefficients of the even powers depend on a different arbitrary constant.
That gives us our two independent solution.

But equally likely (whatever that means) we may find only one series solution this way.

Is there an ingenious way of finding the other solution (can probably derive the other one from the Wronskian in some cases)? And how do we know how many solutions we can find by subbing in a naive power series ansatz?

thanks
sorry if this is the wrong forum.

 

[Ben Niehoff]

Yes, the second linearly-independent solution can be found by power series. I don't remember all the details, but you can re-derive the whole thing by taking your formal power series solution and applying variation of parameters. There's a presentation of this in Arfken, but it is laborious and confusing. In the end you can find a second formula to plug into your equation, that will generate the second solution.