Link to actual problem [7826] \[ \boxed {y^{\prime \prime }+x y^{\prime }+2 y=0} \]
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 &= x \,{\mathrm e}^{-\frac {x^{2}}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2}}{2}} y}{x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= i \sqrt {\pi }\, \sqrt {2}-\pi \,\operatorname {erf}\left (\frac {i \sqrt {2}\, x}{2}\right ) {\mathrm e}^{-\frac {x^{2}}{2}} x\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2}}{2}} y}{i \sqrt {2}\, \sqrt {\pi }\, {\mathrm e}^{\frac {x^{2}}{2}}-\pi \,\operatorname {erf}\left (\frac {i \sqrt {2}\, x}{2}\right ) x}\right ] \\ \end{align*}