Link to actual problem [7555] \[ \boxed {y^{\prime \prime }+x^{6} y^{\prime }+7 x^{5} y=0} \]
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^{7}}{7}} x\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{7}}{7}} y}{x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= 7^{\frac {1}{7}}-\frac {{\mathrm e}^{-\frac {x^{7}}{7}} x^{7} \left (\Gamma \left (\frac {6}{7}\right )-\Gamma \left (\frac {6}{7}, -\frac {x^{7}}{7}\right )\right )}{\left (-x^{7}\right )^{\frac {6}{7}}}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {\left (-x^{7}\right )^{\frac {6}{7}} {\mathrm e}^{\frac {x^{7}}{7}} y}{-\left (-x^{7}\right )^{\frac {6}{7}} 7^{\frac {1}{7}} {\mathrm e}^{\frac {x^{7}}{7}}+x^{7} \left (\Gamma \left (\frac {6}{7}\right )-\Gamma \left (\frac {6}{7}, -\frac {x^{7}}{7}\right )\right )}\right ] \\ \end{align*}