Link to actual problem [1297] \[ \boxed {x^{2} \left (x^{2}+3 x +3\right ) y^{\prime \prime }+x \left (7 x^{2}+8 x +5\right ) y^{\prime }-\left (-9 x^{2}-2 x +1\right ) y=0} \] 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 {{\mathrm e}^{\frac {5 \sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, x}{3}+\sqrt {3}\right )}{3}} \operatorname {HeunG}\left (\frac {-i \sqrt {3}+3}{i \sqrt {3}+3}, 0, 2, 2, -\frac {1}{3}, \frac {-3 \sqrt {3}+23 i}{3 i-3 \sqrt {3}}, -\frac {2 x}{i \sqrt {3}+3}\right ) \left (2 x +3+i \sqrt {3}\right )^{\frac {-3 \sqrt {3}+23 i}{-6 \sqrt {3}+6 i}} \left (-2 x +i \sqrt {3}-3\right )^{\frac {-7 \sqrt {3}+39 i}{6 \sqrt {3}+18 i}}}{x \left (x^{2}+3 x +3\right )^{\frac {4}{3}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {5 \sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, x}{3}+\sqrt {3}\right )}{3}} x \left (x^{2}+3 x +3\right )^{\frac {4}{3}} \left (2 x +3+i \sqrt {3}\right )^{-\frac {4}{3}+\frac {5 i \sqrt {3}}{6}} \left (-2 x +i \sqrt {3}-3\right )^{-\frac {4}{3}-\frac {5 i \sqrt {3}}{6}} y}{\operatorname {HeunG}\left (\frac {-i \sqrt {3}+3}{i \sqrt {3}+3}, 0, 2, 2, -\frac {1}{3}, \frac {-3 \sqrt {3}+23 i}{3 i-3 \sqrt {3}}, -\frac {2 x}{i \sqrt {3}+3}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {{\mathrm e}^{\frac {5 \sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, x}{3}+\sqrt {3}\right )}{3}} \operatorname {HeunG}\left (\frac {-i \sqrt {3}+3}{i \sqrt {3}+3}, \frac {4 \sqrt {3}+4 i}{\sqrt {3}+3 i}, \frac {10}{3}, \frac {10}{3}, \frac {7}{3}, \frac {-3 \sqrt {3}+23 i}{3 i-3 \sqrt {3}}, -\frac {2 x}{i \sqrt {3}+3}\right ) x^{\frac {1}{3}} \left (-2 x +i \sqrt {3}-3\right )^{\frac {-7 \sqrt {3}+39 i}{6 \sqrt {3}+18 i}} \left (2 x +3+i \sqrt {3}\right )^{\frac {-3 \sqrt {3}+23 i}{-6 \sqrt {3}+6 i}}}{\left (x^{2}+3 x +3\right )^{\frac {4}{3}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {5 \sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, x}{3}+\sqrt {3}\right )}{3}} \left (x^{2}+3 x +3\right )^{\frac {4}{3}} \left (-2 x +i \sqrt {3}-3\right )^{-\frac {4}{3}-\frac {5 i \sqrt {3}}{6}} \left (2 x +3+i \sqrt {3}\right )^{-\frac {4}{3}+\frac {5 i \sqrt {3}}{6}} y}{\operatorname {HeunG}\left (\frac {-i \sqrt {3}+3}{i \sqrt {3}+3}, \frac {4 \sqrt {3}+4 i}{\sqrt {3}+3 i}, \frac {10}{3}, \frac {10}{3}, \frac {7}{3}, \frac {-3 \sqrt {3}+23 i}{3 i-3 \sqrt {3}}, -\frac {2 x}{i \sqrt {3}+3}\right ) x^{\frac {1}{3}}}\right ] \\ \end{align*}