Link to actual problem [9460] \[ \boxed {2 y^{\prime \prime } x -\left (x -1\right ) y^{\prime }+a y=0} \]
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 (-a +\frac {1}{2}, \frac {3}{2}, \frac {x}{2}\right ) \sqrt {x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {KummerM}\left (-a +\frac {1}{2}, \frac {3}{2}, \frac {x}{2}\right ) \sqrt {x}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {KummerU}\left (-a +\frac {1}{2}, \frac {3}{2}, \frac {x}{2}\right ) \sqrt {x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {KummerU}\left (-a +\frac {1}{2}, \frac {3}{2}, \frac {x}{2}\right ) \sqrt {x}}\right ] \\ \end{align*}