Link to actual problem [14007] \[ \boxed {x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }+\left (4 x^{2}+5 x \right ) y^{\prime }+\left (x^{2}+1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. Difference not integer"}
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 {\left (x^{2}-2\right )^{{5}/{8}} \left (x +\sqrt {2}\right )^{-\frac {5}{8}+\frac {\sqrt {2}}{2}} {\mathrm e}^{-\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, x}{2}\right )} \operatorname {HeunG}\left (-1, \frac {-4 \sqrt {10}+9 \sqrt {2}+8}{9+4 \sqrt {2}-4 \sqrt {5}}, \frac {\sqrt {5}}{2}-1, \frac {-4 \sqrt {10}-8 \sqrt {2}-\sqrt {5}+2}{18+8 \sqrt {2}-8 \sqrt {5}}, \frac {3}{2}, -\frac {5}{4}+\sqrt {2}, -\frac {\sqrt {2}\, x}{2}\right ) \left (-x +\sqrt {2}\right )^{\frac {\left (16 \sqrt {5}-56\right ) \sqrt {2}+20 \sqrt {5}-77}{72+32 \sqrt {2}-32 \sqrt {5}}}}{\sqrt {x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {x}\, \left (x +\sqrt {2}\right )^{{5}/{8}} \left (x +\sqrt {2}\right )^{-\frac {\sqrt {2}}{2}} {\mathrm e}^{\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, x}{2}\right )} \left (-x +\sqrt {2}\right )^{\frac {5}{8}+\frac {\sqrt {2}}{2}} y}{\left (x^{2}-2\right )^{{5}/{8}} \operatorname {HeunG}\left (-1, \frac {-4 \sqrt {10}+9 \sqrt {2}+8}{9+4 \sqrt {2}-4 \sqrt {5}}, \frac {\sqrt {5}}{2}-1, \frac {-4 \sqrt {10}-8 \sqrt {2}-\sqrt {5}+2}{18+8 \sqrt {2}-8 \sqrt {5}}, \frac {3}{2}, -\frac {5}{4}+\sqrt {2}, -\frac {\sqrt {2}\, x}{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\left (x^{2}-2\right )^{{5}/{8}} \left (x +\sqrt {2}\right )^{-\frac {5}{8}+\frac {\sqrt {2}}{2}} {\mathrm e}^{-\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, x}{2}\right )} \operatorname {HeunG}\left (-1, \frac {-8 \sqrt {10}+18 \sqrt {2}+16}{9+4 \sqrt {2}-4 \sqrt {5}}, \frac {\sqrt {5}}{2}-\frac {3}{2}, \frac {-4 \sqrt {10}-12 \sqrt {2}+3 \sqrt {5}-7}{18+8 \sqrt {2}-8 \sqrt {5}}, \frac {1}{2}, -\frac {5}{4}+\sqrt {2}, -\frac {\sqrt {2}\, x}{2}\right ) \left (-x +\sqrt {2}\right )^{\frac {\left (16 \sqrt {5}-56\right ) \sqrt {2}+20 \sqrt {5}-77}{72+32 \sqrt {2}-32 \sqrt {5}}}}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x \left (x +\sqrt {2}\right )^{{5}/{8}} \left (x +\sqrt {2}\right )^{-\frac {\sqrt {2}}{2}} {\mathrm e}^{\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, x}{2}\right )} \left (-x +\sqrt {2}\right )^{\frac {5}{8}+\frac {\sqrt {2}}{2}} y}{\left (x^{2}-2\right )^{{5}/{8}} \operatorname {HeunG}\left (-1, \frac {-8 \sqrt {10}+18 \sqrt {2}+16}{9+4 \sqrt {2}-4 \sqrt {5}}, \frac {\sqrt {5}}{2}-\frac {3}{2}, \frac {-4 \sqrt {10}-12 \sqrt {2}+3 \sqrt {5}-7}{18+8 \sqrt {2}-8 \sqrt {5}}, \frac {1}{2}, -\frac {5}{4}+\sqrt {2}, -\frac {\sqrt {2}\, x}{2}\right )}\right ] \\ \end{align*}