Link to actual problem [7183] \[ \boxed {y^{\prime \prime }-y^{\prime } x^{2}-x^{3} y=x^{4}+x^{2}} \]
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*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{x -\frac {1}{2} x^{2}} \operatorname {HeunT}\left (2 \,3^{\frac {2}{3}}, -3, -3 \,3^{\frac {1}{3}}, \frac {3^{\frac {2}{3}} x}{3}+\frac {3^{\frac {2}{3}}}{3}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2}}{2}} {\mathrm e}^{-x} y}{\operatorname {HeunT}\left (2 \,3^{\frac {2}{3}}, -3, -3 \,3^{\frac {1}{3}}, \frac {3^{\frac {2}{3}} \left (1+x \right )}{3}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{\frac {x \left (2 x^{2}+3 x -6\right )}{6}} \operatorname {HeunT}\left (2 \,3^{\frac {2}{3}}, 3, -3 \,3^{\frac {1}{3}}, -\frac {3^{\frac {2}{3}} x}{3}-\frac {3^{\frac {2}{3}}}{3}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {1}{3} x^{3}-\frac {1}{2} x^{2}+x} y}{\operatorname {HeunT}\left (2 \,3^{\frac {2}{3}}, 3, -3 \,3^{\frac {1}{3}}, -\frac {3^{\frac {2}{3}} \left (1+x \right )}{3}\right )}\right ] \\ \end{align*}
\begin{align*} \\ \\ \end{align*}