Link to actual problem [1103] \[ \boxed {\left (3 x -1\right ) y^{\prime \prime }-\left (3 x +2\right ) y^{\prime }+\left (6 x -8\right ) y=0} \]
type detected by program
{"unknown"}
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 {x \left (i \sqrt {7}-1\right )}{2}} \left (3 x -1\right )^{2} \operatorname {KummerM}\left (\frac {3}{2}-\frac {5 i \sqrt {7}}{14}, 3, \frac {i \sqrt {7}\, \left (3 x -1\right )}{3}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x \left (i \sqrt {7}-1\right )}{2}} y}{\left (3 x -1\right )^{2} \operatorname {KummerM}\left (\frac {3}{2}-\frac {5 i \sqrt {7}}{14}, 3, \frac {i \sqrt {7}\, \left (3 x -1\right )}{3}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {x \left (i \sqrt {7}-1\right )}{2}} \left (3 x -1\right )^{2} \operatorname {KummerU}\left (\frac {3}{2}-\frac {5 i \sqrt {7}}{14}, 3, \frac {i \sqrt {7}\, \left (3 x -1\right )}{3}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x \left (i \sqrt {7}-1\right )}{2}} y}{\left (3 x -1\right )^{2} \operatorname {KummerU}\left (\frac {3}{2}-\frac {5 i \sqrt {7}}{14}, 3, \frac {i \sqrt {7}\, \left (3 x -1\right )}{3}\right )}\right ] \\ \end{align*}