Link to actual problem [10846] \[ \boxed {y^{\prime \prime }+2 y^{\prime } a x +\left (b \,x^{4}+a^{2} x^{2}+c x +a \right ) y=0} \]
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 &= 0, \underline {\hspace {1.25 ex}}\eta &= x \,{\mathrm e}^{-\frac {x^{2} \left (2 i \sqrt {b}\, x +3 a \right )}{6}} \operatorname {KummerM}\left (\frac {2}{3}+\frac {i c}{6 \sqrt {b}}, \frac {4}{3}, \frac {2 i \sqrt {b}\, x^{3}}{3}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2} \left (2 i \sqrt {b}\, x +3 a \right )}{6}} y}{x \operatorname {KummerM}\left (\frac {i c +4 \sqrt {b}}{6 \sqrt {b}}, \frac {4}{3}, \frac {2 i \sqrt {b}\, x^{3}}{3}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x \,{\mathrm e}^{-\frac {x^{2} \left (2 i \sqrt {b}\, x +3 a \right )}{6}} \operatorname {KummerU}\left (\frac {2}{3}+\frac {i c}{6 \sqrt {b}}, \frac {4}{3}, \frac {2 i \sqrt {b}\, x^{3}}{3}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2} \left (2 i \sqrt {b}\, x +3 a \right )}{6}} y}{x \operatorname {KummerU}\left (\frac {i c +4 \sqrt {b}}{6 \sqrt {b}}, \frac {4}{3}, \frac {2 i \sqrt {b}\, x^{3}}{3}\right )}\right ] \\ \end{align*}