Link to actual problem [9879] \[ \boxed {x^{2} y^{\prime \prime \prime \prime }+8 x y^{\prime \prime \prime }+12 y^{\prime \prime }-\lambda ^{2} y=0} \]
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 &= \frac {\operatorname {BesselJ}\left (2, 2 \sqrt {\lambda }\, \sqrt {x}\right )}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x y}{\operatorname {BesselJ}\left (2, 2 \sqrt {\lambda }\, \sqrt {x}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\operatorname {BesselY}\left (2, 2 \sqrt {\lambda }\, \sqrt {x}\right )}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x y}{\operatorname {BesselY}\left (2, 2 \sqrt {\lambda }\, \sqrt {x}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\operatorname {BesselJ}\left (2, 2 \sqrt {-\lambda }\, \sqrt {x}\right )}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x y}{\operatorname {BesselJ}\left (2, 2 \sqrt {-\lambda }\, \sqrt {x}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\operatorname {BesselY}\left (2, 2 \sqrt {-\lambda }\, \sqrt {x}\right )}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x y}{\operatorname {BesselY}\left (2, 2 \sqrt {-\lambda }\, \sqrt {x}\right )}\right ] \\ \end{align*}