Link to actual problem [9884] \[ \boxed {x^{4} y^{\prime \prime \prime \prime }+4 x^{3} y^{\prime \prime \prime }-\left (4 n^{2}-1\right ) x^{2} y^{\prime \prime }+\left (4 n^{2}-1\right ) x y^{\prime }-4 y x^{4}=0} \]
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 &= \operatorname {BesselJ}\left (n , \left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {2}\, x \right ) \operatorname {BesselJ}\left (n , \left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {2}\, x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {BesselJ}\left (n , \left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {2}\, x \right ) \operatorname {BesselJ}\left (n , \left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {2}\, x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {BesselJ}\left (n , \left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {2}\, x \right ) \operatorname {BesselY}\left (n , \left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {2}\, x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {BesselJ}\left (n , \left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {2}\, x \right ) \operatorname {BesselY}\left (n , \left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {2}\, x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {BesselY}\left (n , \left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {2}\, x \right ) \operatorname {BesselJ}\left (n , \left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {2}\, x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {BesselY}\left (n , \left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {2}\, x \right ) \operatorname {BesselJ}\left (n , \left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {2}\, x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {BesselY}\left (n , \left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {2}\, x \right ) \operatorname {BesselY}\left (n , \left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {2}\, x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {BesselY}\left (n , \left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {2}\, x \right ) \operatorname {BesselY}\left (n , \left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {2}\, x \right )}\right ] \\ \end{align*}