Link to actual problem [9530] \[ \boxed {x^{2} y^{\prime \prime }+x \left (2 x +1\right ) y^{\prime }-4 y=0} \]
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 &= 1-\frac {4 x -3}{2 x^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {2 x^{2} y}{2 x^{2}-4 x +3}\right ] \\ \end{align*}