Link to actual problem [9887] \[ \boxed {x^{4} y^{\prime \prime \prime \prime }+6 x^{3} y^{\prime \prime \prime }+\left (4 x^{4}+\left (-\rho ^{2}-\sigma ^{2}+7\right ) x^{2}\right ) y^{\prime \prime }+\left (16 x^{3}+\left (-\rho ^{2}-\sigma ^{2}+1\right ) x \right ) y^{\prime }+\left (\rho ^{2} \sigma ^{2}+8 x^{2}\right ) y=0} \]
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 &= \operatorname {BesselJ}\left (\frac {\rho }{2}+\frac {\sigma }{2}, x\right ) \operatorname {BesselJ}\left (\frac {\rho }{2}-\frac {\sigma }{2}, x\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {BesselJ}\left (\frac {\rho }{2}+\frac {\sigma }{2}, x\right ) \operatorname {BesselJ}\left (\frac {\rho }{2}-\frac {\sigma }{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 (\frac {\rho }{2}+\frac {\sigma }{2}, x\right ) \operatorname {BesselY}\left (\frac {\rho }{2}-\frac {\sigma }{2}, x\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {BesselJ}\left (\frac {\rho }{2}+\frac {\sigma }{2}, x\right ) \operatorname {BesselY}\left (\frac {\rho }{2}-\frac {\sigma }{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 (\frac {\rho }{2}+\frac {\sigma }{2}, x\right ) \operatorname {BesselJ}\left (\frac {\rho }{2}-\frac {\sigma }{2}, x\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {BesselY}\left (\frac {\rho }{2}+\frac {\sigma }{2}, x\right ) \operatorname {BesselJ}\left (\frac {\rho }{2}-\frac {\sigma }{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 (\frac {\rho }{2}+\frac {\sigma }{2}, x\right ) \operatorname {BesselY}\left (\frac {\rho }{2}-\frac {\sigma }{2}, x\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {BesselY}\left (\frac {\rho }{2}+\frac {\sigma }{2}, x\right ) \operatorname {BesselY}\left (\frac {\rho }{2}-\frac {\sigma }{2}, x\right )}\right ] \\ \end{align*}