Link to actual problem [8089] \[ \boxed {x^{2} \left (x +1\right ) y^{\prime \prime }+x \left (1-10 x \right ) y^{\prime }-\left (9-10 x \right ) y=0} \]
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 &= 572+\frac {715 x^{4}+234 x^{2}+52 x +5}{x^{3}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{3} y}{715 x^{4}+572 x^{3}+234 x^{2}+52 x +5}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {y}{8 x^{10}+91 x^{9}+468 x^{8}+1430 x^{7}+2860 x^{6}+3861 x^{5}+3432 x^{4}+1716 x^{3}}\right ] \\ \end{align*}