Link to actual problem [7556] \[ \boxed {\left (x^{8}+1\right ) y^{\prime \prime }-16 x^{7} y^{\prime }+72 x^{6} y=0} \]
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 &= -\frac {7}{9}+x^{8}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{-\frac {7}{9}+x^{8}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{9}-\frac {9}{7} x\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x^{9}-\frac {9}{7} x}\right ] \\ \end{align*}