Seek the solutions that are power series to the differential equation:
I assume we first need to assume that so that can be differentiated.
I have that
Yes, the question is in the chapter on power series and it explicitly says that the answer should be a power series. The book hasn't covered complex Taylor and Mclaurin series yet, so even if I know that the answer is some branch of -2log(1-z) I can't use that to derive the power series representation of it.
I have that [/QUOTE]
Yes, that derivative, and the change in index, is correct. Now put them into your differential equation:
Now you have to change the index in that second sum. let i= n+1. Then
or, changing the dummy index back to n, [tex]\sum_{n= 1}^\infty n c_n z^n[/itex].
Now, we can write the differential equation as
Since that second sum does not begin until n= 1, we need to do n= 0 separately. If n= 0 then
For n> 0, equating coefficients of the same powers of z,
or
That is, [tex]c_1= 2c_0, , , , etc.