Link to actual problem [11911] \[ \boxed {3 x y^{\prime \prime }-\left (x -2\right ) y^{\prime }-2 y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. 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 &= \operatorname {hypergeom}\left (\left [2\right ], \left [\frac {2}{3}\right ], \frac {x}{3}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {hypergeom}\left (\left [2\right ], \left [\frac {2}{3}\right ], \frac {x}{3}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{\frac {1}{3}} {\mathrm e}^{\frac {x}{3}} \left (x +4\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {x}{3}} y}{x^{\frac {1}{3}} \left (x +4\right )}\right ] \\ \end{align*}