Link to actual problem [7137] \[ \boxed {y^{\prime \prime }-x y^{\prime }-y x=x} \]
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} y}{2+x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (-x -2\right ) \pi \,{\mathrm e}^{-x -2} \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (2+x \right )}{2}\right )+i \sqrt {\pi }\, \sqrt {2}\, {\mathrm e}^{\frac {1}{2} x^{2}+x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{x} y}{-\pi \,{\mathrm e}^{-2} \left (2+x \right ) \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (2+x \right )}{2}\right )+i \sqrt {\pi }\, \sqrt {2}\, {\mathrm e}^{\frac {x^{2}}{2}} {\mathrm e}^{2 x}}\right ] \\ \end{align*}