Link to actual problem [9451] \[ \boxed {y^{\prime \prime } x -\left (x^{2}-x -2\right ) y^{\prime }-x \left (x +3\right ) y=0} \]
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 [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{\frac {x^{2}}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= y \,{\mathrm e}^{-\frac {x^{2}}{2}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{\frac {x^{2}}{2}} \left (\int \frac {{\mathrm e}^{-\frac {1}{2} x^{2}-x}}{x^{2}}d x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {x^{2}}{2}} y}{\int \frac {{\mathrm e}^{-\frac {x^{2}}{2}} {\mathrm e}^{-x}}{x^{2}}d x}\right ] \\ \end{align*}