Link to actual problem [1241] \[ \boxed {y^{\prime \prime }+y^{\prime } x^{5}+6 y x^{4}=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Ordinary point", "second order series method. Taylor series method"}
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^{\frac {1}{6}}-\frac {{\mathrm e}^{-\frac {x^{6}}{6}} x^{6} \left (\Gamma \left (\frac {5}{6}\right )-\Gamma \left (\frac {5}{6}, -\frac {x^{6}}{6}\right )\right )}{\left (-x^{6}\right )^{\frac {5}{6}}}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {\left (-x^{6}\right )^{\frac {5}{6}} {\mathrm e}^{\frac {x^{6}}{6}} y}{-\left (-x^{6}\right )^{\frac {5}{6}} 6^{\frac {1}{6}} {\mathrm e}^{\frac {x^{6}}{6}}+x^{6} \left (\Gamma \left (\frac {5}{6}\right )-\Gamma \left (\frac {5}{6}, -\frac {x^{6}}{6}\right )\right )}\right ] \\ \end{align*}