Link to actual problem [9598] \[ \boxed {\left (2 x^{2}+6 x +4\right ) y^{\prime \prime }+\left (10 x^{2}+21 x +8\right ) y^{\prime }+\left (12 x^{2}+17 x +8\right ) y=0} \]
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 &= {\mathrm e}^{-2 x} \operatorname {HeunC}\left (-1, -\frac {5}{2}, 4, -\frac {7}{4}, \frac {7}{2}, -x -1\right ) \left (2+x \right )^{4}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{2 x} y}{\operatorname {HeunC}\left (-1, -\frac {5}{2}, 4, -\frac {7}{4}, \frac {7}{2}, -x -1\right ) \left (2+x \right )^{4}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-2 x} \operatorname {HeunC}\left (-1, \frac {5}{2}, 4, -\frac {7}{4}, \frac {7}{2}, -x -1\right ) \left (2+x \right )^{4} \left (1+x \right )^{\frac {5}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{2 x} y}{\operatorname {HeunC}\left (-1, \frac {5}{2}, 4, -\frac {7}{4}, \frac {7}{2}, -x -1\right ) \left (2+x \right )^{4} \left (1+x \right )^{\frac {5}{2}}}\right ] \\ \end{align*}