Link to actual problem [7557] \[ \boxed {y^{\prime \prime }+x^{5} y^{\prime }+6 x^{4} y=0} \]
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 &= {\mathrm e}^{-\frac {x^{6}}{6}} x\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{6}}{6}} y}{x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= 6+{\mathrm e}^{-\frac {x^{6}}{6}} \left (-7776 x^{6}\right )^{\frac {1}{6}} \left (\Gamma \left (\frac {5}{6}\right )-\Gamma \left (\frac {5}{6}, -\frac {x^{6}}{6}\right )\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{6}}{6}} y}{\left (-7776 x^{6}\right )^{\frac {1}{6}} \Gamma \left (\frac {5}{6}\right )-\left (-7776 x^{6}\right )^{\frac {1}{6}} \Gamma \left (\frac {5}{6}, -\frac {x^{6}}{6}\right )+6 \,{\mathrm e}^{\frac {x^{6}}{6}}}\right ] \\ \end{align*}