Link to actual problem [11310] \[ \boxed {x^{4} y^{\prime \prime }+2 x^{3} \left (x +1\right ) y^{\prime }+y n^{2}=0} \]
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 &= \frac {\operatorname {HeunD}\left (-8 \left (-n^{2}\right )^{\frac {1}{4}}, -1-\frac {8 \left (i \left (-n^{2}\right )^{\frac {3}{4}}-\sqrt {-n^{2}}\, n \right )}{n}, -\frac {16 i \left (-n^{2}\right )^{\frac {3}{4}}}{n}, 1-\frac {8 \left (i \left (-n^{2}\right )^{\frac {3}{4}}+\sqrt {-n^{2}}\, n \right )}{n}, -\frac {i n -\left (-n^{2}\right )^{\frac {1}{4}} x}{\left (-n^{2}\right )^{\frac {1}{4}} x +i n}\right ) {\mathrm e}^{-\frac {i x \sqrt {-n^{2}}}{n}-\frac {i n}{x}-x}}{\sqrt {x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {x}\, {\mathrm e}^{\frac {i x \sqrt {-n^{2}}}{n}} {\mathrm e}^{\frac {i n}{x}} {\mathrm e}^{x} y}{\operatorname {HeunD}\left (-8 \left (-n^{2}\right )^{\frac {1}{4}}, \frac {-8 i \left (-n^{2}\right )^{\frac {3}{4}}-n +8 \sqrt {-n^{2}}\, n}{n}, -\frac {16 i \left (-n^{2}\right )^{\frac {3}{4}}}{n}, \frac {n -8 i \left (-n^{2}\right )^{\frac {3}{4}}-8 \sqrt {-n^{2}}\, n}{n}, \frac {\left (-n^{2}\right )^{\frac {1}{4}} x -i n}{\left (-n^{2}\right )^{\frac {1}{4}} x +i n}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\operatorname {HeunD}\left (8 \left (-n^{2}\right )^{\frac {1}{4}}, -1-\frac {8 \left (i \left (-n^{2}\right )^{\frac {3}{4}}-\sqrt {-n^{2}}\, n \right )}{n}, -\frac {16 i \left (-n^{2}\right )^{\frac {3}{4}}}{n}, 1-\frac {8 \left (i \left (-n^{2}\right )^{\frac {3}{4}}+\sqrt {-n^{2}}\, n \right )}{n}, -\frac {i n -\left (-n^{2}\right )^{\frac {1}{4}} x}{\left (-n^{2}\right )^{\frac {1}{4}} x +i n}\right ) {\mathrm e}^{\frac {i \sqrt {-n^{2}}\, x^{2}+i n^{2}-n \,x^{2}}{x n}}}{\sqrt {x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {x}\, {\mathrm e}^{-\frac {i x \sqrt {-n^{2}}}{n}} {\mathrm e}^{-\frac {i n}{x}} {\mathrm e}^{x} y}{\operatorname {HeunD}\left (8 \left (-n^{2}\right )^{\frac {1}{4}}, \frac {-8 i \left (-n^{2}\right )^{\frac {3}{4}}-n +8 \sqrt {-n^{2}}\, n}{n}, -\frac {16 i \left (-n^{2}\right )^{\frac {3}{4}}}{n}, \frac {n -8 i \left (-n^{2}\right )^{\frac {3}{4}}-8 \sqrt {-n^{2}}\, n}{n}, \frac {\left (-n^{2}\right )^{\frac {1}{4}} x -i n}{\left (-n^{2}\right )^{\frac {1}{4}} x +i n}\right )}\right ] \\ \end{align*}