Link to actual problem [7549] \[ \boxed {\left (2 x^{2}-8 x +11\right ) y^{\prime \prime }-16 \left (x -2\right ) y^{\prime }+36 y=0} \]
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 {31}{5}+x^{3}-6 x^{2}+\frac {111}{10} x\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{-\frac {31}{5}+x^{3}-6 x^{2}+\frac {111}{10} x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= -\frac {16577}{8}+\frac {x \left (4 x^{5}-48 x^{4}+330 x^{3}-5445 x +13068\right )}{4}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x^{6}-12 x^{5}+\frac {165}{2} x^{4}-\frac {16577}{8}-\frac {5445}{4} x^{2}+3267 x}\right ] \\ \end{align*}