Link to actual problem [8112] \[ \boxed {y^{\prime \prime }+\left (t^{2}+2 t +1\right ) y^{\prime }-\left (4 t +4\right ) y=0} \]
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 [R &= t, S \left (R \right ) &= \frac {y}{t^{4}+4 t^{3}+6 t^{2}+8 t +5}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (t +1\right ) \left (t^{3}+3 t^{2}+3 t +5\right ) \left (\int \frac {{\mathrm e}^{-\frac {1}{3} t^{3}-t^{2}-t}}{\left (t +1\right )^{2} \left (t^{3}+3 t^{2}+3 t +5\right )^{2}}d t \right )\right ] \\ \left [R &= t, S \left (R \right ) &= \frac {y}{\left (t +1\right ) \left (t^{3}+3 t^{2}+3 t +5\right ) \left (\int \frac {{\mathrm e}^{-\frac {t^{3}}{3}} {\mathrm e}^{-t^{2}} {\mathrm e}^{-t}}{\left (t +1\right )^{2} \left (t^{3}+3 t^{2}+3 t +5\right )^{2}}d t \right )}\right ] \\ \end{align*}