Link to actual problem [8183] \[ \boxed {2 x^{2} y^{\prime \prime }-\left (3 x +2\right ) y^{\prime }+\frac {\left (2 x -1\right ) y}{x}=0} \]
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 {5 x +2}{\sqrt {x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {x}\, y}{5 x +2}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {1}{x}} \operatorname {hypergeom}\left (\left [2\right ], \left [-\frac {1}{2}\right ], \frac {1}{x}\right ) x^{2}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {1}{x}} y}{\operatorname {hypergeom}\left (\left [2\right ], \left [-\frac {1}{2}\right ], \frac {1}{x}\right ) x^{2}}\right ] \\ \end{align*}