Link to actual problem [9443] \[ \boxed {y^{\prime \prime } x -2 \left (x -1\right ) y^{\prime }-y=0} \]
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 [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-x} y}{\operatorname {BesselI}\left (0, x\right )-\operatorname {BesselI}\left (1, x\right )}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= -\frac {{\mathrm e}^{-x} y}{\operatorname {BesselK}\left (0, -x \right )-\operatorname {BesselK}\left (1, -x \right )}\right ] \\ \end{align*}