Link to actual problem [7512] \[ \boxed {2 y^{\prime \prime }+x y^{\prime }+3 y=0} \]
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 &= {\mathrm e}^{-\frac {x^{2}}{4}} \left (x^{2}-2\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2}}{4}} y}{x^{2}-2}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (-x^{2}+2\right ) \sqrt {\pi }\, \operatorname {erfi}\left (\frac {x}{2}\right ) {\mathrm e}^{-\frac {x^{2}}{4}}+2 x\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2}}{4}} y}{2 x \,{\mathrm e}^{\frac {x^{2}}{4}}-\sqrt {\pi }\, \left (x^{2}-2\right ) \operatorname {erfi}\left (\frac {x}{2}\right )}\right ] \\ \end{align*}