I'm trying to solve the following ODE:
$$ y''+ \left( \frac{7/2}{x+1} + \frac{7/2}{x-1} + \frac{3}{x+\frac{1}{\sqrt{2}}} \right) y'+\left( \frac{\frac{7}{8}(3 \sqrt{2}-5)}{x-1} + \frac{\frac{7}{8}(3 \sqrt{2}+5)}{x+1} + \frac{-21/(2\sqrt{2})}{x+\sqrt{2}} \right) y=0 $$
Clearly, there's four regular singularities, but each coefficient present only shares two of the same at a time. For four regular singularities, Heun's ODE is the candidate to map onto, but I'm having trouble finding a mobius transformation that helps me get to it. Perhaps my question comes down to whether or not it's possible for a mapping to exist such that ${-1,1,\sqrt{2},1/\sqrt{2}} \rightarrow \{0,1,a, \infty\}$, in no particular order, such that we obtain a Heun ODE or a confluent Heun ODE.
Some really impressive work has been done before in mapping the Heun ODE on to particular examples of ODE's of form
$$ f'' + \left( \sum_{i}^{4} \ \frac{\alpha_i}{x-\tilde{x}_i} \right) f' + \left( \sum_{i}^{4} \ \frac{\beta}{x-\tilde{x}_i} \right) f = 0 $$
I've tried to employ a similar method to what I have, but I keep getting expressions too nasty for me or Mathematica to simplify, though I'm still trying.
