Link to actual problem [7611] \[ \boxed {x^{2} \left (x^{2}-2 x +1\right ) y^{\prime \prime }-x \left (x +3\right ) y^{\prime }+\left (4+x \right ) y=0} \]
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 &= \frac {x^{2} {\mathrm e}^{-\frac {4}{-1+x}}}{-1+x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (-1+x \right ) {\mathrm e}^{\frac {4}{-1+x}} y}{x^{2}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {x^{2} \operatorname {expIntegral}_{1}\left (-\frac {4 x}{-1+x}\right ) {\mathrm e}^{-\frac {4 x}{-1+x}}}{-1+x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (-1+x \right ) {\mathrm e}^{\frac {4 x}{-1+x}} y}{x^{2} \operatorname {expIntegral}_{1}\left (-\frac {4 x}{-1+x}\right )}\right ] \\ \end{align*}