I am trying to solve the following equation;
$$
U''+\left( \frac{1}{t}+\frac{3}{t-1}\right)U'+\left(\frac{1}{t}+C\right)\frac{U}{t(t-1)}=0
$$
where U is a function of t and C is constant.
The above equation is similar to a form of Riemann equation.


Could anyone please provide any support on how the above equation can be solved. I am trying to obtain the solution in form of hyper-geometric series.



Solution of second order differential equation with singularities at 0,1, and ∞
I'm not sure if this is more useful than the *Maple* solution given earlier, but Wolfram *Mathematica* finds a slightly different solution in terms of associated Legendre functions:


$$U(t) = \frac{1}{1-t} \left( k\_1 P\_\ell^2(2t-1) + k\_2 Q\_\ell^2(2t-1) \right)$$


Here, $k\_1$ and $k\_2$ are constants of integration, and $P\_\ell^m$ and $Q\_\ell^m$ are the associated Legendre functions of the [first](http://mathworld.wolfram.com/LegendreFunctionoftheFirstKind.html) and [second](http://mathworld.wolfram.com/LegendreFunctionoftheSecondKind.html) kinds, with $\ell = \frac{1}{2} \left(\sqrt{9-4 C}-1\right)$.


