Link to actual problem [8241] \[ \boxed {y^{\prime \prime }-2 x y^{\prime }+8 y=0} \]
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 {y}{4 x^{4}-12 x^{2}+3}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{x^{2}} \left (-4 x^{3}+10 x \right )+\operatorname {erfi}\left (x \right ) \sqrt {\pi }\, \left (4 x^{4}-12 x^{2}+3\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{{\mathrm e}^{x^{2}} \left (-4 x^{3}+10 x \right )+4 \left (x^{4}-3 x^{2}+\frac {3}{4}\right ) \sqrt {\pi }\, \operatorname {erfi}\left (x \right )}\right ] \\ \end{align*}