Link to actual problem [4716] \[ \boxed {\left (-x^{2}+x \right ) y^{\prime \prime }-y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. Difference is integer"}
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 &= \operatorname {hypergeom}\left (\left [\frac {3}{2}-\frac {i \sqrt {3}}{2}, \frac {1}{2}-\frac {i \sqrt {3}}{2}\right ], \left [1-i \sqrt {3}\right ], \frac {1}{x}\right ) \left (x^{\frac {1}{2}+\frac {i \sqrt {3}}{2}}-x^{-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{-\frac {i \sqrt {3}}{2}} \sqrt {x}\, y}{\operatorname {hypergeom}\left (\left [\frac {3}{2}-\frac {i \sqrt {3}}{2}, \frac {1}{2}-\frac {i \sqrt {3}}{2}\right ], \left [1-i \sqrt {3}\right ], \frac {1}{x}\right ) \left (-1+x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {hypergeom}\left (\left [\frac {i \sqrt {3}}{2}+\frac {3}{2}, \frac {1}{2}+\frac {i \sqrt {3}}{2}\right ], \left [1+i \sqrt {3}\right ], \frac {1}{x}\right ) \left (x^{\frac {1}{2}-\frac {i \sqrt {3}}{2}}-x^{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {i \sqrt {3}}{2}} \sqrt {x}\, y}{\operatorname {hypergeom}\left (\left [\frac {i \sqrt {3}}{2}+\frac {3}{2}, \frac {1}{2}+\frac {i \sqrt {3}}{2}\right ], \left [1+i \sqrt {3}\right ], \frac {1}{x}\right ) \left (-1+x \right )}\right ] \\ \end{align*}