OK, then I read Frobenius method in mathworld (I learned when I took ODE 2):
<http://mathworld.wolfram.com/FrobeniusMethod.html>


My question is:
Are there any ODEs where the solution is given by full Laurent series?, i.e its negative indexed coeffecients are not zero starting from some negative integer.


Thanks in advance.



Laurent series expansion for ODE.
Every analytic function in a ring has a Laurent expansion. Thus if your differential equation
has a solution analytic in a ring it has a Laurent expansion. See the examples given in other answers. 


The difference between a full Laurent
series (I understand that "full" means infinitely many coefficients in both directions) and
a Frobenius series (which is infinite in only one direction) is that you cannot manipulate
with a full Laurent series formally. In particular, you cannot multiply two such series:
the coefficient of the product is a series, rather than a finite sum.


For this reason, the use of full Laurent series is limited.
The advantage of the one-suded series is that you can substitute a formal one-sided series to the differential equation and effectively determine its coefficients (like in Frobenius method). You cannot do this
with a full Laurent series.


Thus a full Laurent series solution (when exists) is usually impossible to find explicitly.


