Link to actual problem [7178] \[ \boxed {y^{\prime \prime }-y^{\prime } x^{2}-y x^{2}=x^{2}} \]
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 &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {HeunT}\left (3^{\frac {2}{3}}, 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 (3^{\frac {2}{3}}, 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 (3^{\frac {2}{3}}, -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}^{-\frac {x^{3}}{3}} {\mathrm e}^{-x} y}{\operatorname {HeunT}\left (3^{\frac {2}{3}}, -3, 2 \,3^{\frac {1}{3}}, -\frac {3^{\frac {2}{3}} x}{3}\right )}\right ] \\ \end{align*}