Link to actual problem [9462] \[ \boxed {\left (2 x -1\right ) y^{\prime \prime }-\left (3 x -4\right ) y^{\prime }+\left (-3+x \right ) y=0} \]
type detected by program
{"kovacic"}
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 &= \frac {{\mathrm e}^{\frac {x}{2}} \operatorname {KummerM}\left (1, \frac {3}{4}, -\frac {1}{4}+\frac {x}{2}\right )}{\left (2 x -1\right )^{{1}/{4}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {x}{2}} \left (2 x -1\right )^{{1}/{4}} y}{\operatorname {KummerM}\left (1, \frac {3}{4}, -\frac {1}{4}+\frac {x}{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {{\mathrm e}^{\frac {x}{2}} \operatorname {KummerU}\left (1, \frac {3}{4}, -\frac {1}{4}+\frac {x}{2}\right )}{\left (2 x -1\right )^{{1}/{4}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {x}{2}} \left (2 x -1\right )^{{1}/{4}} y}{\operatorname {KummerU}\left (1, \frac {3}{4}, -\frac {1}{4}+\frac {x}{2}\right )}\right ] \\ \end{align*}