Link to actual problem [8127] \[ \boxed {4 z y^{\prime \prime }+2 \left (1-z \right ) y^{\prime }-y=0} \]
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 &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{\frac {z}{2}}\right ] \\ \left [R &= z, S \left (R \right ) &= {\mathrm e}^{-\frac {z}{2}} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{\frac {z}{2}} \operatorname {erf}\left (\frac {\sqrt {2}\, \sqrt {z}}{2}\right )\right ] \\ \left [R &= z, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {z}{2}} y}{\operatorname {erf}\left (\frac {\sqrt {2}\, \sqrt {z}}{2}\right )}\right ] \\ \end{align*}