Link to actual problem [1303] \[ \boxed {x \left (x^{2}+x +3\right ) y^{\prime \prime }+\left (-x^{2}+x +4\right ) y^{\prime }+y x=0} \] 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 {11}\, \arctan \left (\frac {\left (2 x +1\right ) \sqrt {11}}{11}\right )}{66}} \operatorname {HeunG}\left (\frac {\sqrt {11}+i}{i-\sqrt {11}}, \frac {255 i \sqrt {11}+699}{163 i \sqrt {11}+517}, \frac {5}{6}+\frac {5 i \sqrt {11}}{66}, \frac {20 \sqrt {11}+370 i}{59 \sqrt {11}+385 i}, \frac {2}{3}, -\frac {7}{6}+\frac {5 i \sqrt {11}}{66}, -\frac {2 x}{1+i \sqrt {11}}\right ) \left (x^{2}+x +3\right )^{\frac {7}{12}} \left (2 x +i \sqrt {11}+1\right )^{-\frac {7}{12}+\frac {5 i \sqrt {11}}{132}} \left (-2 x +i \sqrt {11}-1\right )^{\frac {3085 \sqrt {11}+6911 i}{2040 \sqrt {11}+3828 i}}}{x^{\frac {1}{3}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {5 \sqrt {11}\, \arctan \left (\frac {\left (2 x +1\right ) \sqrt {11}}{11}\right )}{66}} x^{\frac {1}{3}} \left (2 x +i \sqrt {11}+1\right )^{\frac {7}{12}} \left (2 x +i \sqrt {11}+1\right )^{-\frac {5 i \sqrt {11}}{132}} \left (-2 x +i \sqrt {11}-1\right )^{-\frac {5 i \sqrt {11}}{132}-\frac {19}{12}} y}{\operatorname {HeunG}\left (\frac {\sqrt {11}+i}{i-\sqrt {11}}, \frac {255 i \sqrt {11}+699}{163 i \sqrt {11}+517}, \frac {5}{6}+\frac {5 i \sqrt {11}}{66}, \frac {20 \sqrt {11}+370 i}{59 \sqrt {11}+385 i}, \frac {2}{3}, -\frac {7}{6}+\frac {5 i \sqrt {11}}{66}, -\frac {2 x}{1+i \sqrt {11}}\right ) \left (x^{2}+x +3\right )^{\frac {7}{12}}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {5 \sqrt {11}\, \arctan \left (\frac {\left (2 x +1\right ) \sqrt {11}}{11}\right )}{66}} \operatorname {HeunG}\left (\frac {\sqrt {11}+i}{i-\sqrt {11}}, \frac {5864 i \sqrt {11}+55424}{1341 i \sqrt {11}+19701}, \frac {7}{6}+\frac {5 i \sqrt {11}}{66}, \frac {119 \sqrt {11}+1495 i}{177 \sqrt {11}+1155 i}, \frac {4}{3}, -\frac {7}{6}+\frac {5 i \sqrt {11}}{66}, -\frac {2 x}{1+i \sqrt {11}}\right ) \left (x^{2}+x +3\right )^{\frac {7}{12}} \left (2 x +i \sqrt {11}+1\right )^{-\frac {7}{12}+\frac {5 i \sqrt {11}}{132}} \left (-2 x +i \sqrt {11}-1\right )^{\frac {3085 \sqrt {11}+6911 i}{2040 \sqrt {11}+3828 i}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {5 \sqrt {11}\, \arctan \left (\frac {\left (2 x +1\right ) \sqrt {11}}{11}\right )}{66}} \left (2 x +i \sqrt {11}+1\right )^{\frac {7}{12}} \left (2 x +i \sqrt {11}+1\right )^{-\frac {5 i \sqrt {11}}{132}} \left (-2 x +i \sqrt {11}-1\right )^{-\frac {5 i \sqrt {11}}{132}-\frac {19}{12}} y}{\operatorname {HeunG}\left (\frac {\sqrt {11}+i}{i-\sqrt {11}}, \frac {5864 i \sqrt {11}+55424}{1341 i \sqrt {11}+19701}, \frac {7}{6}+\frac {5 i \sqrt {11}}{66}, \frac {119 \sqrt {11}+1495 i}{177 \sqrt {11}+1155 i}, \frac {4}{3}, -\frac {7}{6}+\frac {5 i \sqrt {11}}{66}, -\frac {2 x}{1+i \sqrt {11}}\right ) \left (x^{2}+x +3\right )^{\frac {7}{12}}}\right ] \\ \end{align*}