Link to actual problem [9705] \[ \boxed {y^{\prime \prime }+\frac {2 \left (x^{2}-1\right ) y^{\prime }}{x \left (x -1\right )^{2}}+\frac {\left (-2 x^{2}+2 x +2\right ) y}{x^{2} \left (x -1\right )^{2}}=0} \]
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^{2}}{-1+x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (-1+x \right ) y}{x^{2}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {x \left (2 \ln \left (x \right ) x^{2}-2 \ln \left (-1+x \right ) x^{2}-2 x \ln \left (x \right )+2 \ln \left (-1+x \right ) x -2 x +1\right )}{2 \left (-1+x \right )^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (-1+x \right )^{2} y}{x \left (\left (-x^{2}+x \right ) \ln \left (-1+x \right )+\left (x^{2}-x \right ) \ln \left (x \right )-x +\frac {1}{2}\right )}\right ] \\ \end{align*}