Link to actual problem [7627] \[ \boxed {x^{2} \left (1-2 x \right ) y^{\prime \prime }-x \left (4 x +5\right ) y^{\prime }+\left (9+4 x \right ) y=0} \]
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 (8 x +1\right )}{\left (2 x -1\right )^{6}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (2 x -1\right )^{6} y}{8 x^{4}+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 (-4096 x^{4}+16384 x^{3}+24576 x \ln \left (x \right )-36864 x^{2}+3072 \ln \left (x \right )-4872 x +9375\right )}{4096 \left (2 x -1\right )^{6}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (2 x -1\right )^{6} y}{\left (\left (-6 x -\frac {3}{4}\right ) \ln \left (x \right )+x^{4}-4 x^{3}+9 x^{2}+\frac {609 x}{512}-\frac {9375}{4096}\right ) x^{3}}\right ] \\ \end{align*}