Link to actual problem [2921] \[ \boxed {x^{2} y^{\prime \prime }+x \left (1-x \right ) y^{\prime }-7 y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. Difference not 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 &= \sqrt {x}\, {\mathrm e}^{\frac {x}{2}} \left (\operatorname {BesselI}\left (-\frac {1}{2}+\sqrt {7}, -\frac {x}{2}\right )+\operatorname {BesselI}\left (\frac {1}{2}+\sqrt {7}, -\frac {x}{2}\right )\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {x}{2}} y}{\sqrt {x}\, \left (\operatorname {BesselI}\left (-\frac {1}{2}+\sqrt {7}, -\frac {x}{2}\right )+\operatorname {BesselI}\left (\frac {1}{2}+\sqrt {7}, -\frac {x}{2}\right )\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= -\sqrt {x}\, {\mathrm e}^{\frac {x}{2}} \left (-\operatorname {BesselK}\left (-\frac {1}{2}+\sqrt {7}, -\frac {x}{2}\right )+\operatorname {BesselK}\left (\frac {1}{2}+\sqrt {7}, -\frac {x}{2}\right )\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {x}{2}} y}{\sqrt {x}\, \left (\operatorname {BesselK}\left (-\frac {1}{2}+\sqrt {7}, -\frac {x}{2}\right )-\operatorname {BesselK}\left (\frac {1}{2}+\sqrt {7}, -\frac {x}{2}\right )\right )}\right ] \\ \end{align*}