Link to actual problem [9389] \[ \boxed {y^{\prime \prime }-x^{2} y^{\prime }-\left (x +1\right )^{2} y=0} \]
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 &= \operatorname {HeunT}\left (0, -3, 2 \,3^{\frac {1}{3}}, \frac {3^{\frac {2}{3}} x}{3}\right ) {\mathrm e}^{-x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{x} y}{\operatorname {HeunT}\left (0, -3, 2 \,3^{\frac {1}{3}}, \frac {3^{\frac {2}{3}} x}{3}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {HeunT}\left (0, 3, 2 \,3^{\frac {1}{3}}, -\frac {3^{\frac {2}{3}} x}{3}\right ) {\mathrm e}^{\frac {1}{3} x^{3}+x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-x} {\mathrm e}^{-\frac {x^{3}}{3}} y}{\operatorname {HeunT}\left (0, 3, 2 \,3^{\frac {1}{3}}, -\frac {3^{\frac {2}{3}} x}{3}\right )}\right ] \\ \end{align*}