Link to actual problem [9538] \[ \boxed {x^{2} y^{\prime \prime }+\left (x^{2}+2\right ) x y^{\prime }+\left (x^{2}-2\right ) y=0} \]
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 [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {{\mathrm e}^{-\frac {x^{2}}{2}}}{x^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{\frac {x^{2}}{2}} x^{2} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {i \sqrt {2}\, \sqrt {\pi }\, {\mathrm e}^{-\frac {x^{2}}{2}} \operatorname {erf}\left (\frac {i \sqrt {2}\, x}{2}\right )+2 x}{x^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2}}{2}} x^{2} y}{i \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {i \sqrt {2}\, x}{2}\right )+2 x \,{\mathrm e}^{\frac {x^{2}}{2}}}\right ] \\ \end{align*}