Link to actual problem [9844] \[ \boxed {2 \left (x -\operatorname {a1} \right ) \left (x -\operatorname {a2} \right ) \left (x -\operatorname {a3} \right ) y^{\prime \prime \prime }+\left (9 x^{2}-6 \left (\operatorname {a1} +\operatorname {a2} +\operatorname {a3} \right ) x +3 \operatorname {a1} \operatorname {a2} +3 \operatorname {a1} \operatorname {a3} +3 \operatorname {a2} \operatorname {a3} \right ) y^{\prime \prime }-2 \left (\left (n^{2}+n -3\right ) x +b \right ) y^{\prime }-n \left (1+n \right ) y=0} \]
type detected by program
{"unknown"}
type detected by Maple
[[_3rd_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 &= \operatorname {HeunG}\left (\frac {\operatorname {a1} -\operatorname {a3}}{\operatorname {a2} -\operatorname {a3}}, -\frac {-\operatorname {a3} \,n^{2}-\operatorname {a3} n +\operatorname {a1} +\operatorname {a2} +\operatorname {a3} -b}{4 \left (\operatorname {a2} -\operatorname {a3} \right )}, -\frac {n}{2}, \frac {n}{2}+\frac {1}{2}, \frac {1}{2}, \frac {1}{2}, -\frac {-x +\operatorname {a3}}{\operatorname {a2} -\operatorname {a3}}\right )^{2}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{{\operatorname {HeunG}\left (\frac {\operatorname {a1} -\operatorname {a3}}{\operatorname {a2} -\operatorname {a3}}, \frac {\left (n^{2}+n -1\right ) \operatorname {a3} +b -\operatorname {a1} -\operatorname {a2}}{4 \operatorname {a2} -4 \operatorname {a3}}, -\frac {n}{2}, \frac {n}{2}+\frac {1}{2}, \frac {1}{2}, \frac {1}{2}, \frac {x -\operatorname {a3}}{\operatorname {a2} -\operatorname {a3}}\right )}^{2}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {HeunG}\left (\frac {\operatorname {a1} -\operatorname {a3}}{\operatorname {a2} -\operatorname {a3}}, \frac {\operatorname {a3} \,n^{2}+\operatorname {a3} n -3 \operatorname {a3} +b}{4 \operatorname {a2} -4 \operatorname {a3}}, \frac {n}{2}+1, -\frac {n}{2}+\frac {1}{2}, \frac {3}{2}, \frac {1}{2}, -\frac {-x +\operatorname {a3}}{\operatorname {a2} -\operatorname {a3}}\right )^{2} \left (-x +\operatorname {a3} \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{{\operatorname {HeunG}\left (\frac {\operatorname {a1} -\operatorname {a3}}{\operatorname {a2} -\operatorname {a3}}, \frac {\left (n^{2}+n -3\right ) \operatorname {a3} +b}{4 \operatorname {a2} -4 \operatorname {a3}}, \frac {n}{2}+1, -\frac {n}{2}+\frac {1}{2}, \frac {3}{2}, \frac {1}{2}, \frac {x -\operatorname {a3}}{\operatorname {a2} -\operatorname {a3}}\right )}^{2} \left (-x +\operatorname {a3} \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {HeunG}\left (\frac {\operatorname {a1} -\operatorname {a3}}{\operatorname {a2} -\operatorname {a3}}, -\frac {-\operatorname {a3} \,n^{2}-\operatorname {a3} n +\operatorname {a1} +\operatorname {a2} +\operatorname {a3} -b}{4 \left (\operatorname {a2} -\operatorname {a3} \right )}, -\frac {n}{2}, \frac {n}{2}+\frac {1}{2}, \frac {1}{2}, \frac {1}{2}, -\frac {-x +\operatorname {a3}}{\operatorname {a2} -\operatorname {a3}}\right ) \operatorname {HeunG}\left (\frac {\operatorname {a1} -\operatorname {a3}}{\operatorname {a2} -\operatorname {a3}}, \frac {\operatorname {a3} \,n^{2}+\operatorname {a3} n -3 \operatorname {a3} +b}{4 \operatorname {a2} -4 \operatorname {a3}}, \frac {n}{2}+1, -\frac {n}{2}+\frac {1}{2}, \frac {3}{2}, \frac {1}{2}, -\frac {-x +\operatorname {a3}}{\operatorname {a2} -\operatorname {a3}}\right ) \sqrt {-x +\operatorname {a3}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {HeunG}\left (\frac {\operatorname {a1} -\operatorname {a3}}{\operatorname {a2} -\operatorname {a3}}, \frac {\left (n^{2}+n -1\right ) \operatorname {a3} +b -\operatorname {a1} -\operatorname {a2}}{4 \operatorname {a2} -4 \operatorname {a3}}, -\frac {n}{2}, \frac {n}{2}+\frac {1}{2}, \frac {1}{2}, \frac {1}{2}, \frac {x -\operatorname {a3}}{\operatorname {a2} -\operatorname {a3}}\right ) \operatorname {HeunG}\left (\frac {\operatorname {a1} -\operatorname {a3}}{\operatorname {a2} -\operatorname {a3}}, \frac {\left (n^{2}+n -3\right ) \operatorname {a3} +b}{4 \operatorname {a2} -4 \operatorname {a3}}, \frac {n}{2}+1, -\frac {n}{2}+\frac {1}{2}, \frac {3}{2}, \frac {1}{2}, \frac {x -\operatorname {a3}}{\operatorname {a2} -\operatorname {a3}}\right ) \sqrt {-x +\operatorname {a3}}}\right ] \\ \end{align*}