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I have a problem with finding the closed form solution of the following ODEs. Closed form here means that the solution can be presented as integrals/ power series. Here is the ODE : I only consider $x\in (0,1)$ and $c_i$ are known non-zero real numbers.

$\Large{y''(x) + [\frac{c_1}{x^2}+\frac{c_2}{(1-x)^2} +c_3(\frac{1}{x}+\frac{1}{1-x})]y'(x)+[c_4(\frac{1}{x}+\frac{1}{1-x})+\frac{c_5}{x^2}+\frac{c_6}{(1-x)^2}]y(x)=0} $

I can find the solution inform of power series which is very ugly. I would like to ask you all that whether you can give a method that can be used to find the solution in a nicer way. Thanks so much for your time. I really appreciate it.

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If using power series, it is natural to substitute $x=t+1/2$, so that the expansion is centered at $t=0$ and the formulas are more symmetric. This might decrease the ugliness a little. // I'm not optimistic about the closed form solution. When I include only some of the terms in the coefficients of $y'$ and $y$, Maple gives solutions as combinations of increasingly obscure special functions. When I put all the terms in, it gives up. – user53153 Feb 18 '13 at 0:13
Thanks Gerry for your suggestion !!! – steven Feb 18 '13 at 1:09

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