Link to actual problem [9883] \[ \boxed {x^{4} y^{\prime \prime \prime \prime }-2 n \left (1+n \right ) x^{2} y^{\prime \prime }+4 n \left (1+n \right ) x y^{\prime }+\left (a \,x^{4}+n \left (1+n \right ) \left (n +3\right ) \left (n -2\right )\right ) y=0} \]
type detected by program
{"unknown"}
type detected by Maple
[[_high_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 &= \sqrt {x}\, \operatorname {BesselJ}\left (n +\frac {1}{2}, \left (-a \right )^{{1}/{4}} x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\sqrt {x}\, \operatorname {BesselJ}\left (n +\frac {1}{2}, \left (-a \right )^{{1}/{4}} x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \sqrt {x}\, \operatorname {BesselY}\left (n +\frac {1}{2}, \left (-a \right )^{{1}/{4}} x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\sqrt {x}\, \operatorname {BesselY}\left (n +\frac {1}{2}, \left (-a \right )^{{1}/{4}} x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \sqrt {x}\, \operatorname {BesselJ}\left (n +\frac {1}{2}, \sqrt {-\sqrt {-a}}\, x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\sqrt {x}\, \operatorname {BesselJ}\left (n +\frac {1}{2}, \sqrt {-\sqrt {-a}}\, x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \sqrt {x}\, \operatorname {BesselY}\left (n +\frac {1}{2}, \sqrt {-\sqrt {-a}}\, x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\sqrt {x}\, \operatorname {BesselY}\left (n +\frac {1}{2}, \sqrt {-\sqrt {-a}}\, x \right )}\right ] \\ \end{align*}