Link to actual problem [9680] \[ \boxed {y^{\prime \prime }-\frac {\left (2 x^{2}-1\right ) y^{\prime }}{x^{3}}+\frac {y}{x^{4}}=0} \]
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^{3}+2 x -\frac {1}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x y}{x^{4}+2 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^{3}-2 \sqrt {2}\, x +\frac {\sqrt {2}}{x}\right ) \sqrt {\pi }\, \operatorname {erfi}\left (\frac {\sqrt {2}}{2 x}\right )+\left (2 x^{2}-2\right ) {\mathrm e}^{\frac {1}{2 x^{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x y}{-\operatorname {erfi}\left (\frac {\sqrt {2}}{2 x}\right ) \left (x^{4}+2 x^{2}-1\right ) \sqrt {\pi }\, \sqrt {2}+\left (2 x^{3}-2 x \right ) {\mathrm e}^{\frac {1}{2 x^{2}}}}\right ] \\ \end{align*}