Link to actual problem [8052] \[ \boxed {x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-3 x \left (-x^{2}+1\right ) y^{\prime }+4 y=0} \]
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^{2}}{\left (x^{2}+1\right )^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x^{2}+1\right )^{2} y}{x^{2}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {x^{2} \left (x^{2}+2 \ln \left (x \right )\right )}{2 \left (x^{2}+1\right )^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {2 \left (x^{2}+1\right )^{2} y}{x^{2} \left (x^{2}+2 \ln \left (x \right )\right )}\right ] \\ \end{align*}