Link to actual problem [7551] \[ \boxed {\left (x^{2}-8 x +14\right ) y^{\prime \prime }-8 \left (x -4\right ) y^{\prime }+20 y=0} \]
type detected by program
{"kovacic"}
type detected by Maple
[[_2nd_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 &= \frac {14}{5}+\frac {x \left (x -8\right )}{5}, \underline {\hspace {1.25 ex}}\eta &= y x\right ] \\ \left [R &= \frac {y \,{\mathrm e}^{10 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (x -4\right ) \sqrt {2}}{2}\right )}}{\left (x^{2}-8 x +14\right )^{\frac {5}{2}}}, S \left (R \right ) &= -\frac {5 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 x -8\right ) \sqrt {2}}{4}\right )}{2}\right ] \\ \end{align*}