Link to actual problem [14807] \[ \boxed {9 x y^{\prime \prime }+14 y^{\prime }+y \left (x -1\right )=0} \] 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 &= {\mathrm e}^{-\frac {i x}{3}} \operatorname {KummerM}\left (\frac {7}{9}-\frac {i}{6}, \frac {14}{9}, \frac {2 i x}{3}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {i x}{3}} y}{\operatorname {KummerM}\left (\frac {7}{9}-\frac {i}{6}, \frac {14}{9}, \frac {2 i x}{3}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {i x}{3}} \operatorname {KummerU}\left (\frac {7}{9}-\frac {i}{6}, \frac {14}{9}, \frac {2 i x}{3}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {i x}{3}} y}{\operatorname {KummerU}\left (\frac {7}{9}-\frac {i}{6}, \frac {14}{9}, \frac {2 i x}{3}\right )}\right ] \\ \end{align*}