Link to actual problem [9634] \[ \boxed {x^{3} y^{\prime \prime }+x \left (x +1\right ) y^{\prime }-2 y=0} \]
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 {{\mathrm e}^{\frac {1}{x}} \left (1+x \right )}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x \,{\mathrm e}^{-\frac {1}{x}} y}{1+x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= 1-\frac {{\mathrm e}^{\frac {1}{x}} \left (1+x \right ) \operatorname {expIntegral}_{1}\left (\frac {1}{x}\right )}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x y}{-{\mathrm e}^{\frac {1}{x}} \left (1+x \right ) \operatorname {expIntegral}_{1}\left (\frac {1}{x}\right )+x}\right ] \\ \end{align*}