Link to actual problem [9824] \[ \boxed {x^{2} y^{\prime \prime \prime }-2 \left (1+n \right ) x y^{\prime \prime }+\left (a \,x^{2}+6 n \right ) y^{\prime }-2 y a x=0} \]
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*} \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x^{2} a +4 n -2}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{n +\frac {1}{2}} \operatorname {BesselJ}\left (-n -\frac {1}{2}, \sqrt {a}\, x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{-n} y}{\sqrt {x}\, \operatorname {BesselJ}\left (-n -\frac {1}{2}, \sqrt {a}\, x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{n +\frac {1}{2}} \operatorname {BesselY}\left (-n -\frac {1}{2}, \sqrt {a}\, x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{-n} y}{\sqrt {x}\, \operatorname {BesselY}\left (-n -\frac {1}{2}, \sqrt {a}\, x \right )}\right ] \\ \end{align*}