Link to actual problem [9826] \[ \boxed {x^{2} y^{\prime \prime \prime }-\left (x +\nu \right ) x y^{\prime \prime }+\nu \left (1+2 x \right ) y^{\prime }-\nu \left (x +1\right ) y=0} \]
type detected by program
{"unknown"}
type detected by Maple
[[_3rd_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*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{\frac {\nu }{2}+\frac {1}{2}} \operatorname {BesselJ}\left (-\nu -1, 2 \sqrt {\nu }\, \sqrt {x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{-\frac {\nu }{2}} y}{\sqrt {x}\, \operatorname {BesselJ}\left (-\nu -1, 2 \sqrt {\nu }\, \sqrt {x}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{\frac {\nu }{2}+\frac {1}{2}} \operatorname {BesselY}\left (-\nu -1, 2 \sqrt {\nu }\, \sqrt {x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{-\frac {\nu }{2}} y}{\sqrt {x}\, \operatorname {BesselY}\left (-\nu -1, 2 \sqrt {\nu }\, \sqrt {x}\right )}\right ] \\ \end{align*}