Link to actual problem [8188] \[ \boxed {4 y^{\prime \prime }+\frac {3 \left (-x^{2}+2\right ) y}{\left (-x^{2}+1\right )^{2}}=0} \]
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 &= \left (x^{2}-1\right )^{\frac {3}{4}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\left (x^{2}-1\right )^{\frac {3}{4}}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (x^{2}-1\right )^{\frac {1}{4}} x\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\left (x^{2}-1\right )^{\frac {1}{4}} x}\right ] \\ \end{align*}