Link to actual problem [8040] \[ \boxed {x^{2} \left (2 x +1\right ) y^{\prime \prime }+x \left (3 x^{2}+14 x +5\right ) y^{\prime }+\left (12 x^{2}+18 x +4\right ) y=0} \]
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 {3 x}{2}} \operatorname {HeunC}\left (-\frac {3}{4}, \frac {1}{4}, 0, \frac {21}{32}, -\frac {5}{32}, 2 x +1\right )}{x^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {3 x}{2}} x^{2} y}{\operatorname {HeunC}\left (-\frac {3}{4}, \frac {1}{4}, 0, \frac {21}{32}, -\frac {5}{32}, 2 x +1\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {{\mathrm e}^{-\frac {3 x}{2}} \operatorname {HeunC}\left (-\frac {3}{4}, -\frac {1}{4}, 0, \frac {21}{32}, -\frac {5}{32}, 2 x +1\right )}{\left (2 x +1\right )^{\frac {1}{4}} x^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {3 x}{2}} \left (2 x +1\right )^{\frac {1}{4}} x^{2} y}{\operatorname {HeunC}\left (-\frac {3}{4}, -\frac {1}{4}, 0, \frac {21}{32}, -\frac {5}{32}, 2 x +1\right )}\right ] \\ \end{align*}