Link to actual problem [7502] \[ \boxed {t y^{\prime \prime }+\left (t^{2}-1\right ) y^{\prime }+y t^{2}=0} \]
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}^{t -\frac {1}{2} t^{2}} \left (-1+t \right )\right ] \\ \left [R &= t, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {t^{2}}{2}} {\mathrm e}^{-t} y}{-1+t}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= -{\mathrm e}^{t -\frac {1}{2} t^{2}} \left (-\operatorname {hypergeom}\left (\left [\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \frac {1}{2} t^{2}-2 t +2\right ) t +\operatorname {hypergeom}\left (\left [-\frac {1}{2}\right ], \left [\frac {1}{2}\right ], \frac {1}{2} t^{2}-2 t +2\right )+2 \operatorname {hypergeom}\left (\left [\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \frac {1}{2} t^{2}-2 t +2\right )\right )\right ] \\ \left [R &= t, S \left (R \right ) &= \frac {{\mathrm e}^{-t} {\mathrm e}^{\frac {t^{2}}{2}} y}{-\operatorname {hypergeom}\left (\left [-\frac {1}{2}\right ], \left [\frac {1}{2}\right ], \frac {\left (t -2\right )^{2}}{2}\right )+\operatorname {hypergeom}\left (\left [\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \frac {\left (t -2\right )^{2}}{2}\right ) \left (t -2\right )}\right ] \\ \end{align*}