Let’s write the equation in the form

(1)

That is an linear DE and its general solution is in the form

(2)

where and are two independently linear solutions of the (1) and two constants. Now we will research a solution of (1) analytic in , so that il can be written as

(3)

The way for finding the is to substitute the (3) into the (1) obtaining

(4)

The term is pratically the constant in (2). Observing the (4) it is easy to see that is and the same holds for all the of odd index. For the of even index we have

(5)

The recursive relations (5) permit us to arrive to the Taylor expansion around of the analytic solutions of (1)

(6)

That is about the analytic solutions of the form . For the non analytic solutions of the form the problem is a little more difficult…

Kind regards