Link to actual problem [11077] \[ \boxed {x^{2} \left (a^{2} x^{2 n}-1\right ) y^{\prime \prime }+x \left (a p \,x^{n}+q \right ) y^{\prime }+\left (a r \,x^{n}+s \right ) y=0} \]
type detected by program
{"unknown"}
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 &= x^{\frac {\sqrt {q^{2}+2 q +4 s +1}}{2}+\frac {q}{2}+\frac {1}{2}} \operatorname {HeunG}\left (-1, \frac {-\left (q^{2}+2 q +4 s +1\right )^{\frac {3}{2}}+\left (-p^{2}+\left (2 n -q -2\right ) p +q^{2}+2 q +4 s +1\right ) \sqrt {q^{2}+2 q +4 s +1}+\left (-q -1\right ) p^{2}+\left (-q^{2}+\left (-3+2 n \right ) q +2 n -2 r -2\right ) p +4 n r -2 q r -4 r}{2 n^{2} \left (2 n -2-p -q \right )}, \frac {\sqrt {q^{2}+2 q +4 s +1}+q -1}{2 n}, \frac {\sqrt {q^{2}+2 q +4 s +1}+q +1}{2 n}, 1+\frac {\sqrt {q^{2}+2 q +4 s +1}}{n}, -\frac {p -q}{2 n}, -a \,x^{n}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{-\frac {\sqrt {q^{2}+2 q +4 s +1}}{2}} x^{-\frac {q}{2}} y}{\sqrt {x}\, \operatorname {HeunG}\left (-1, \frac {-\left (q^{2}+2 q +4 s +1\right )^{\frac {3}{2}}+\left (-p^{2}+\left (2 n -q -2\right ) p +q^{2}+2 q +4 s +1\right ) \sqrt {q^{2}+2 q +4 s +1}+\left (-q -1\right ) p^{2}+\left (-q^{2}+\left (-3+2 n \right ) q +2 n -2 r -2\right ) p +4 n r -2 q r -4 r}{2 n^{2} \left (2 n -2-p -q \right )}, \frac {\sqrt {q^{2}+2 q +4 s +1}+q -1}{2 n}, \frac {\sqrt {q^{2}+2 q +4 s +1}+q +1}{2 n}, \frac {n +\sqrt {q^{2}+2 q +4 s +1}}{n}, -\frac {p -q}{2 n}, -a \,x^{n}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{\frac {q}{2}+\frac {1}{2}-\frac {\sqrt {q^{2}+2 q +4 s +1}}{2}} \operatorname {HeunG}\left (-1, \frac {-\left (q^{2}+2 q +4 s +1\right )^{\frac {3}{2}}+\left (p^{2}+\left (-2 n +q +2\right ) p +q^{2}+2 q +4 s +1\right ) \sqrt {q^{2}+2 q +4 s +1}+\left (-q -1\right ) p^{2}+\left (-q^{2}+\left (-3+2 n \right ) q +2 n -2 r -2\right ) p +4 n r -2 q r -4 r}{2 n^{2} \left (2 n -2-p -q \right )}, -\frac {\sqrt {q^{2}+2 q +4 s +1}-q -1}{2 n}, -\frac {\sqrt {q^{2}+2 q +4 s +1}-q +1}{2 n}, 1-\frac {\sqrt {q^{2}+2 q +4 s +1}}{n}, -\frac {p -q}{2 n}, -a \,x^{n}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{-\frac {q}{2}} x^{\frac {\sqrt {q^{2}+2 q +4 s +1}}{2}} y}{\sqrt {x}\, \operatorname {HeunG}\left (-1, \frac {-\left (q^{2}+2 q +4 s +1\right )^{\frac {3}{2}}+\left (p^{2}+\left (-2 n +q +2\right ) p +q^{2}+2 q +4 s +1\right ) \sqrt {q^{2}+2 q +4 s +1}+\left (-q -1\right ) p^{2}+\left (-q^{2}+\left (-3+2 n \right ) q +2 n -2 r -2\right ) p +4 n r -2 q r -4 r}{2 n^{2} \left (2 n -2-p -q \right )}, -\frac {\sqrt {q^{2}+2 q +4 s +1}-q -1}{2 n}, -\frac {\sqrt {q^{2}+2 q +4 s +1}-q +1}{2 n}, \frac {n -\sqrt {q^{2}+2 q +4 s +1}}{n}, -\frac {p -q}{2 n}, -a \,x^{n}\right )}\right ] \\ \end{align*}