Link to actual problem [8164] \[ \boxed {4 x^{2} y^{\prime \prime }+4 x \left (1-x \right ) y^{\prime }+\left (2 x -9\right ) y=0} \]
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 &= \frac {x^{2}+2 x +2}{x^{\frac {3}{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {3}{2}} y}{x^{2}+2 x +2}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {{\mathrm e}^{x}}{x^{\frac {3}{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= x^{\frac {3}{2}} {\mathrm e}^{-x} y\right ] \\ \end{align*}