Link to actual problem [8173] \[ \boxed {x y^{\prime \prime }+\left (x +n \right ) y^{\prime }+\left (n +1\right ) y=0} \]
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 [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{x} y}{\operatorname {hypergeom}\left (\left [-1\right ], \left [n \right ], x\right )}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {x^{n} {\mathrm e}^{x} y}{x \operatorname {hypergeom}\left (\left [-n \right ], \left [-n +2\right ], x\right )}\right ] \\ \end{align*}