Link to actual problem [8174] \[ \boxed {x^{4} y^{\prime \prime }+x y^{\prime }+y=0} \]
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 -\frac {1}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x y}{x^{2}-1}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (\sqrt {2}\, x -\frac {\sqrt {2}}{x}\right ) \sqrt {\pi }\, \operatorname {erfi}\left (\frac {\sqrt {2}}{2 x}\right )+2 \,{\mathrm e}^{\frac {1}{2 x^{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x y}{\sqrt {\pi }\, \sqrt {2}\, \operatorname {erfi}\left (\frac {\sqrt {2}}{2 x}\right ) \left (-1+x \right ) \left (1+x \right )+2 x \,{\mathrm e}^{\frac {1}{2 x^{2}}}}\right ] \\ \end{align*}