Link to actual problem [7641] \[ \boxed {x^{2} \left (1-x \right ) y^{\prime \prime }-x \left (3-5 x \right ) y^{\prime }+\left (4-5 x \right ) y=0} \]
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 [R &= x, S \left (R \right ) &= \frac {y}{x^{2} \left (-1+x \right )^{3}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{2} \left (-\left (-1+x \right )^{3} \ln \left (-1+x \right )+\left (-1+x \right )^{3} \ln \left (x \right )-x^{2}+\frac {5 x}{2}-\frac {11}{6}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x^{2} \left (-\left (-1+x \right )^{3} \ln \left (-1+x \right )+\left (-1+x \right )^{3} \ln \left (x \right )-x^{2}+\frac {5 x}{2}-\frac {11}{6}\right )}\right ] \\ \end{align*}