Link to actual problem [7576] \[ \boxed {18 x^{2} \left (x +1\right ) y^{\prime \prime }+3 x \left (x^{2}+11 x +5\right ) y^{\prime }-\left (-5 x^{2}-2 x +1\right ) y=0} \]
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 {{\mathrm e}^{-\frac {x}{6}} \operatorname {HeunC}\left (\frac {1}{6}, -\frac {1}{2}, -\frac {1}{6}, -\frac {5}{36}, \frac {1}{4}, -x \right )}{x^{\frac {1}{6}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x}{6}} x^{\frac {1}{6}} y}{\operatorname {HeunC}\left (\frac {1}{6}, -\frac {1}{2}, -\frac {1}{6}, -\frac {5}{36}, \frac {1}{4}, -x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {x}{6}} \operatorname {HeunC}\left (\frac {1}{6}, \frac {1}{2}, -\frac {1}{6}, -\frac {5}{36}, \frac {1}{4}, -x \right ) x^{\frac {1}{3}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x}{6}} y}{\operatorname {HeunC}\left (\frac {1}{6}, \frac {1}{2}, -\frac {1}{6}, -\frac {5}{36}, \frac {1}{4}, -x \right ) x^{\frac {1}{3}}}\right ] \\ \end{align*}