Link to actual problem [7578] \[ \boxed {2 x^{2} y^{\prime \prime }+x \left (5+x \right ) y^{\prime }-\left (2-3 x \right ) y=0} \]
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 &= \sqrt {x}\, {\mathrm e}^{-\frac {x}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x}{2}} y}{\sqrt {x}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= 2+\frac {i {\mathrm e}^{-\frac {x}{2}} x^{\frac {5}{2}} \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {i \sqrt {2}\, \sqrt {x}}{2}\right )+2 x +6}{x^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{2} {\mathrm e}^{\frac {x}{2}} y}{i \sqrt {\pi }\, x^{\frac {5}{2}} \operatorname {erf}\left (\frac {i \sqrt {2}\, \sqrt {x}}{2}\right ) \sqrt {2}+2 \,{\mathrm e}^{\frac {x}{2}} \left (x^{2}+x +3\right )}\right ] \\ \end{align*}