Link to actual problem [9812] \[ \boxed {\left (2 x -1\right ) y^{\prime \prime \prime }-8 y^{\prime } x +8 y=0} \]
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*} \\ \left [R &= x, S \left (R \right ) &= -\frac {y}{x}\right ] \\ \end{align*}
\begin{align*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= -\frac {x \,{\mathrm e}^{-1} \operatorname {Ei}_{1}\left (2 x -1\right )}{2}+\frac {\operatorname {Ei}_{1}\left (-2+4 x \right ) {\mathrm e}^{2 x -2}}{4}+\frac {{\mathrm e}^{-2 x}}{4}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {4 \,{\mathrm e}^{2 x} y}{{\mathrm e}^{4 x} {\mathrm e}^{-2} \operatorname {Ei}_{1}\left (-2+4 x \right )-2 x \,{\mathrm e}^{-1} \operatorname {Ei}_{1}\left (2 x -1\right ) {\mathrm e}^{2 x}+1}\right ] \\ \end{align*}