Link to actual problem [9880] \[ \boxed {x^{2} y^{\prime \prime \prime \prime }+\left (2 n -2 \nu +4\right ) x y^{\prime \prime \prime }+\left (n -\nu +1\right ) \left (n -\nu +2\right ) y^{\prime \prime }-\frac {b^{4} y}{16}=0} \]
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 &= x^{-\frac {n}{2}+\frac {\nu }{2}} \operatorname {BesselI}\left (n -\nu , b \sqrt {x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {n}{2}} x^{-\frac {\nu }{2}} y}{\operatorname {BesselI}\left (n -\nu , b \sqrt {x}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{-\frac {n}{2}+\frac {\nu }{2}} \operatorname {BesselJ}\left (n -\nu , b \sqrt {x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {n}{2}} x^{-\frac {\nu }{2}} y}{\operatorname {BesselJ}\left (n -\nu , b \sqrt {x}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{-\frac {n}{2}+\frac {\nu }{2}} \operatorname {BesselK}\left (n -\nu , b \sqrt {x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {n}{2}} x^{-\frac {\nu }{2}} y}{\operatorname {BesselK}\left (n -\nu , b \sqrt {x}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{-\frac {n}{2}+\frac {\nu }{2}} \operatorname {BesselY}\left (n -\nu , b \sqrt {x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {n}{2}} x^{-\frac {\nu }{2}} y}{\operatorname {BesselY}\left (n -\nu , b \sqrt {x}\right )}\right ] \\ \end{align*}