Link to actual problem [8045] \[ \boxed {x^{2} \left (1-2 x \right ) y^{\prime \prime }-x \left (5-4 x \right ) y^{\prime }+\left (9-4 x \right ) y=0} \]
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 &= \frac {x^{3}}{\left (2 x -1\right )^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (2 x -1\right )^{2} y}{x^{3}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= -\frac {x^{3} \left (-2 x +\ln \left (x \right )\right )}{\left (2 x -1\right )^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (2 x -1\right )^{2} y}{x^{3} \left (2 x -\ln \left (x \right )\right )}\right ] \\ \end{align*}