Link to actual problem [9806] \[ \boxed {x y^{\prime \prime \prime }-\left (x +2 v \right ) y^{\prime \prime }-\left (x -2 v -1\right ) y^{\prime }+\left (x -1\right ) y=0} \]
type detected by program
{"unknown"}
type detected by Maple
[[_3rd_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*} \\ \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {x^{-v} y}{x \operatorname {BesselI}\left (-v -1, x\right )}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {x^{-v} y}{x \operatorname {BesselK}\left (v +1, x\right )}\right ] \\ \end{align*}