Link to actual problem [2918] \[ \boxed {x^{2} y^{\prime \prime }+\frac {x y^{\prime }}{\left (-x^{2}+1\right )^{2}}+y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. Complex roots"}
type detected by Maple
[[_2nd_order, _with_linear_symmetries]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{i} \left (x^{2}-1\right )^{\frac {1}{4}-\frac {i}{2}+\frac {i \sqrt {3}}{4}} \operatorname {HeunC}\left (\frac {1}{2}, -\frac {i \sqrt {3}}{2}, i, -\frac {5}{8}, -\frac {3}{8}, -\frac {1}{x^{2}-1}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{-i} \left (x^{2}-1\right )^{-\frac {1}{4}+\frac {i}{2}} \left (x^{2}-1\right )^{-\frac {i \sqrt {3}}{4}} y}{\operatorname {HeunC}\left (\frac {1}{2}, -\frac {i \sqrt {3}}{2}, i, -\frac {5}{8}, -\frac {3}{8}, -\frac {1}{x^{2}-1}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{i} \operatorname {HeunC}\left (\frac {1}{2}, \frac {i \sqrt {3}}{2}, i, -\frac {5}{8}, -\frac {3}{8}, -\frac {1}{x^{2}-1}\right ) \left (x^{2}-1\right )^{-\frac {i \sqrt {3}}{4}+\frac {1}{4}-\frac {i}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{-i} \left (x^{2}-1\right )^{\frac {i \sqrt {3}}{4}} \left (x^{2}-1\right )^{-\frac {1}{4}+\frac {i}{2}} y}{\operatorname {HeunC}\left (\frac {1}{2}, \frac {i \sqrt {3}}{2}, i, -\frac {5}{8}, -\frac {3}{8}, -\frac {1}{x^{2}-1}\right )}\right ] \\ \end{align*}