Link to actual problem [8171] \[ \boxed {2 x y^{\prime \prime }+5 \left (1-2 x \right ) y^{\prime }-5 y=0} \]
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 {10 x +1}{x^{\frac {3}{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {3}{2}} y}{10 x +1}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (\frac {10 \sqrt {5}}{\sqrt {x}}+\frac {\sqrt {5}}{x^{\frac {3}{2}}}\right ) \sqrt {\pi }\, \operatorname {erfi}\left (\sqrt {5}\, \sqrt {x}\right )-\frac {10 \,{\mathrm e}^{5 x}}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {5}{2}} y}{10 x \sqrt {5}\, \sqrt {\pi }\, \left (x +\frac {1}{10}\right ) \operatorname {erfi}\left (\sqrt {5}\, \sqrt {x}\right )-10 \,{\mathrm e}^{5 x} x^{\frac {3}{2}}}\right ] \\ \end{align*}