Link to actual problem [8097] \[ \boxed {x^{2} y^{\prime \prime }+x \left (-2 x^{2}+1\right ) y^{\prime }-4 \left (2 x^{2}+1\right ) y=0} \]
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 {{\mathrm e}^{-x^{2}} y}{x^{2}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= -1+\frac {x^{4} {\mathrm e}^{x^{2}} \operatorname {expIntegral}_{1}\left (x^{2}\right )+1}{x^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{2} y}{x^{4} {\mathrm e}^{x^{2}} \operatorname {expIntegral}_{1}\left (x^{2}\right )-x^{2}+1}\right ] \\ \end{align*}