5.3.1.2 Step 2

Recall that from step 1 we have found \(\left [ \sqrt {r}\right ] _{c}\) and its associated \(\alpha _{c}^{+},\alpha _{c}^{-}\) (this is done for each pole of \(r\)) and we have found \(\left [ \sqrt {r}\right ] _{\infty }\) and its associated \(\alpha _{\infty }^{+},\alpha _{\infty }^{-}\). From these we now found a possible \(d\) values and trying each \(d\geq 0\). The value of \(d\) is found using the following for each combination of \(s\left ( c\right ) \) where \(s\left ( c\right ) \) is \(+\) or \(-\)

\[ d=\alpha _{\infty }^{\pm }-\sum _{c}\alpha _{c}^{\pm }\]

We keep only the non negative values of \(d\). It is important to note that we have to find an integer positive value for \(d\) to continue. If no such value is found from the above, then we stop here as this means no Liouvillian solution exist using case 1. Then we go to case two or case three if it is available.

If we do find \(d\geq 0\), then we now find corresponding candidate \(\omega _{d}\) using

\begin{align*} \omega _{d} & =\sum _{c}\left ( (\pm ) \left [ \sqrt {r}\right ] _{c}+\frac {\alpha _{c}^{\pm }}{x-c}\right ) + (\pm ) \left [ \sqrt {r}\right ] _{\infty }\end{align*}