Link to actual problem [9467] \[ \boxed {4 x y^{\prime \prime }+4 m y^{\prime }-\left (x -2 m -4 n \right ) y=0} \]
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}{2}} \operatorname {KummerM}\left (-n , m , x\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x}{2}} y}{\operatorname {KummerM}\left (-n , m , x\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {x}{2}} \operatorname {KummerU}\left (-n , m , x\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x}{2}} y}{\operatorname {KummerU}\left (-n , m , x\right )}\right ] \\ \end{align*}