Link to actual problem [7574] \[ \boxed {12 x^{2} \left (x +1\right ) y^{\prime \prime }+x \left (3 x^{2}+35 x +11\right ) y^{\prime }-\left (-5 x^{2}-10 x +1\right ) y=0} \]
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}{4}} \operatorname {HeunC}\left (\frac {1}{4}, -\frac {7}{12}, -\frac {3}{4}, -\frac {1}{12}, \frac {1}{2}, -x \right )}{\left (1+x \right )^{\frac {3}{4}} x^{\frac {1}{4}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x}{4}} \left (1+x \right )^{\frac {3}{4}} x^{\frac {1}{4}} y}{\operatorname {HeunC}\left (\frac {1}{4}, -\frac {7}{12}, -\frac {3}{4}, -\frac {1}{12}, \frac {1}{2}, -x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {{\mathrm e}^{-\frac {x}{4}} \operatorname {HeunC}\left (\frac {1}{4}, \frac {7}{12}, -\frac {3}{4}, -\frac {1}{12}, \frac {1}{2}, -x \right ) x^{\frac {1}{3}}}{\left (1+x \right )^{\frac {3}{4}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x}{4}} \left (1+x \right )^{\frac {3}{4}} y}{\operatorname {HeunC}\left (\frac {1}{4}, \frac {7}{12}, -\frac {3}{4}, -\frac {1}{12}, \frac {1}{2}, -x \right ) x^{\frac {1}{3}}}\right ] \\ \end{align*}