Link to actual problem [7096] \[ \boxed {y^{\prime \prime }+\frac {\left (t^{2}-1\right ) y^{\prime }}{t}+\frac {t^{2} y}{\left (1+{\mathrm e}^{\frac {t^{2}}{2}}\right )^{2}}=0} \]
type detected by program
{"unknown"}
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 &= \frac {1+{\mathrm e}^{\frac {t^{2}}{2}}}{t}, \underline {\hspace {1.25 ex}}\eta &= 0\right ] \\ \left [R &= y, S \left (R \right ) &= -\ln \left (1+{\mathrm e}^{\frac {t^{2}}{2}}\right )+\ln \left ({\mathrm e}^{\frac {t^{2}}{2}}\right )\right ] \\ \end{align*}