Link to actual problem [8075] \[ \boxed {9 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+3 x \left (13 x^{2}+7 x +1\right ) y^{\prime }+\left (25 x^{2}+4 x +1\right ) y=0} \]
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 {x^{\frac {1}{3}}}{x^{2}+x +1}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x^{2}+x +1\right ) y}{x^{\frac {1}{3}}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {x^{\frac {1}{3}} \ln \left (x \right )}{x^{2}+x +1}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x^{2}+x +1\right ) y}{x^{\frac {1}{3}} \ln \left (x \right )}\right ] \\ \end{align*}