Link to actual problem [8270] \[ \boxed {4 \left (t^{2}-3 t +2\right ) y^{\prime \prime }-2 y^{\prime }+y=0} \]
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 &= \sqrt {-1+t}\right ] \\ \left [R &= t, S \left (R \right ) &= \frac {y}{\sqrt {-1+t}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= -\frac {\ln \left (t -\frac {3}{2}+\sqrt {t^{2}-3 t +2}\right ) \sqrt {t^{2}-3 t +2}-2 t +4}{2 \sqrt {t -2}}\right ] \\ \left [R &= t, S \left (R \right ) &= \frac {2 \sqrt {t -2}\, y}{-\ln \left (t -\frac {3}{2}+\sqrt {t^{2}-3 t +2}\right ) \sqrt {t^{2}-3 t +2}+2 t -4}\right ] \\ \end{align*}