Link to actual problem [8159] \[ \boxed {x^{2} y^{\prime \prime }+x \left (1-x \right ) y^{\prime }-y=0} \]
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 {1}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x y}{1+x}\right ] \\ \end{align*}