Link to actual problem [7566] \[ \boxed {y^{\prime \prime }+3 x y^{\prime }+\left (2 x^{2}+4\right ) y=0} \]
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}{\left (-1+x \right ) \left (1+x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (\sqrt {2}\, x^{2}-\sqrt {2}\right ) \sqrt {\pi }\, {\mathrm e}^{-x^{2}} \operatorname {erfi}\left (\frac {\sqrt {2}\, x}{2}\right )-2 x \,{\mathrm e}^{-\frac {x^{2}}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{x^{2}} {\mathrm e}^{\frac {x^{2}}{2}} y}{\sqrt {2}\, \sqrt {\pi }\, \operatorname {erfi}\left (\frac {\sqrt {2}\, x}{2}\right ) \left (-1+x \right ) \left (1+x \right ) {\mathrm e}^{\frac {x^{2}}{2}}-2 x \,{\mathrm e}^{x^{2}}}\right ] \\ \end{align*}