Link to actual problem [8128] \[ \boxed {f^{\prime \prime }+2 \left (z -1\right ) f^{\prime }+4 f=0} \]
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 [R &= z, S \left (R \right ) &= \frac {{\mathrm e}^{z^{2}} {\mathrm e}^{-2 z} f}{z -1}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= i-\sqrt {\pi }\, \operatorname {erf}\left (i z -i\right ) {\mathrm e}^{-z^{2}+2 z -1} \left (z -1\right )\right ] \\ \left [R &= z, S \left (R \right ) &= \frac {{\mathrm e}^{z^{2}} f}{-\sqrt {\pi }\, {\mathrm e}^{2 z} {\mathrm e}^{-1} \left (z -1\right ) \operatorname {erf}\left (i \left (z -1\right )\right )+i {\mathrm e}^{z^{2}}}\right ] \\ \end{align*}