Link to actual problem [7563] \[ \boxed {x^{2} y^{\prime \prime }-\left (6-7 x \right ) y^{\prime }+8 y=0} \]
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 {6}{x}} \left (-2+x \right )}{x^{5}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{5} {\mathrm e}^{\frac {6}{x}} y}{-2+x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (\frac {108}{x^{4}}-\frac {216}{x^{5}}\right ) {\mathrm e}^{-\frac {6}{x}} \operatorname {expIntegral}_{1}\left (-\frac {6}{x}\right )+\frac {1}{x^{2}}+\frac {12}{x^{3}}-\frac {36}{x^{4}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{5} {\mathrm e}^{\frac {6}{x}} y}{108 \,\operatorname {expIntegral}_{1}\left (-\frac {6}{x}\right ) \left (-2+x \right )+x \,{\mathrm e}^{\frac {6}{x}} \left (x^{2}+12 x -36\right )}\right ] \\ \end{align*}