Link to actual problem [8042] \[ \boxed {9 x^{2} \left (x +1\right ) y^{\prime \prime }+3 x \left (-x^{2}+11 x +5\right ) y^{\prime }+\left (-7 x^{2}+16 x +1\right ) y=0} \]
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 {\operatorname {HeunC}\left (-\frac {1}{3}, \frac {4}{3}, 0, -\frac {1}{9}, \frac {11}{18}, 1+x \right )}{x^{\frac {1}{3}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {1}{3}} y}{\operatorname {HeunC}\left (-\frac {1}{3}, \frac {4}{3}, 0, -\frac {1}{9}, \frac {11}{18}, 1+x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\operatorname {HeunC}\left (-\frac {1}{3}, -\frac {4}{3}, 0, -\frac {1}{9}, \frac {11}{18}, 1+x \right )}{x^{\frac {1}{3}} \left (1+x \right )^{\frac {4}{3}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {1}{3}} \left (1+x \right )^{\frac {4}{3}} y}{\operatorname {HeunC}\left (-\frac {1}{3}, -\frac {4}{3}, 0, -\frac {1}{9}, \frac {11}{18}, 1+x \right )}\right ] \\ \end{align*}