Link to actual problem [12189] \[ \boxed {\left (x^{2}-1\right ) y^{\prime \prime }-6 y=1} \]
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*} \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x^{3}-x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= y +\frac {1}{6}\right ] \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= -1-\frac {3 x \left (\ln \left (1+x \right ) x^{2}-\ln \left (-1+x \right ) x^{2}-\ln \left (1+x \right )+\ln \left (-1+x \right )-2 x \right )}{4}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\frac {\left (3 x^{3}-3 x \right ) \ln \left (-1+x \right )}{4}+\frac {\left (-3 x^{3}+3 x \right ) \ln \left (1+x \right )}{4}+\frac {3 x^{2}}{2}-1}\right ] \\ \end{align*}