Link to actual problem [9444] \[ \boxed {y^{\prime \prime } x -\left (3 x -2\right ) y^{\prime }-\left (2 x -3\right ) y=0} \]
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 &= \operatorname {KummerM}\left (1-\frac {6 \sqrt {17}}{17}, 2, \sqrt {17}\, x \right ) {\mathrm e}^{-\frac {x \left (-3+\sqrt {17}\right )}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x \left (-3+\sqrt {17}\right )}{2}} y}{\operatorname {KummerM}\left (1-\frac {6 \sqrt {17}}{17}, 2, \sqrt {17}\, x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {KummerU}\left (1-\frac {6 \sqrt {17}}{17}, 2, \sqrt {17}\, x \right ) {\mathrm e}^{-\frac {x \left (-3+\sqrt {17}\right )}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x \left (-3+\sqrt {17}\right )}{2}} y}{\operatorname {KummerU}\left (1-\frac {6 \sqrt {17}}{17}, 2, \sqrt {17}\, x \right )}\right ] \\ \end{align*}