Link to actual problem [2912] \[ \boxed {x y^{\prime \prime }-\left (x -1\right ) y^{\prime }-y x=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. Repeated root"}
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 {KummerM}\left (\frac {1}{2}-\frac {\sqrt {5}}{10}, 1, \sqrt {5}\, x \right ) {\mathrm e}^{-\frac {x \left (\sqrt {5}-1\right )}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x \left (\sqrt {5}-1\right )}{2}} y}{\operatorname {KummerM}\left (\frac {1}{2}-\frac {\sqrt {5}}{10}, 1, \sqrt {5}\, x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {KummerU}\left (\frac {1}{2}-\frac {\sqrt {5}}{10}, 1, \sqrt {5}\, x \right ) {\mathrm e}^{-\frac {x \left (\sqrt {5}-1\right )}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x \left (\sqrt {5}-1\right )}{2}} y}{\operatorname {KummerU}\left (\frac {1}{2}-\frac {\sqrt {5}}{10}, 1, \sqrt {5}\, x \right )}\right ] \\ \end{align*}