Link to actual problem [2400] \[ \boxed {x^{3} \left (x^{2}+3\right ) y^{\prime \prime }+5 y^{\prime } x -y \left (x +1\right )=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Irregular singular point"}
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 i \sqrt {3}}{18}+\frac {5 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, x}{3}\right )}{18}} \left (-x +i \sqrt {3}\right )^{1-\frac {5 i \sqrt {3}}{36}} \left (x +i \sqrt {3}\right )^{1+\frac {5 i \sqrt {3}}{36}} \operatorname {HeunC}\left (\frac {10 i \sqrt {3}}{9}, 1+\frac {5 i \sqrt {3}}{18}, 1-\frac {5 i \sqrt {3}}{18}, -\frac {8 i \sqrt {3}}{9}, \frac {4 i \sqrt {3}}{9}+\frac {37}{72}, \frac {1}{2}+\frac {i \sqrt {3}}{2 x}\right )}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x \,{\mathrm e}^{-\frac {5 i \sqrt {3}}{18}} {\mathrm e}^{-\frac {5 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, x}{3}\right )}{18}} \left (-x +i \sqrt {3}\right )^{\frac {5 i \sqrt {3}}{36}} \left (x +i \sqrt {3}\right )^{-\frac {5 i \sqrt {3}}{36}} y}{\left (-x +i \sqrt {3}\right ) \left (x +i \sqrt {3}\right ) \operatorname {HeunC}\left (\frac {10 i \sqrt {3}}{9}, 1+\frac {5 i \sqrt {3}}{18}, 1-\frac {5 i \sqrt {3}}{18}, -\frac {8 i \sqrt {3}}{9}, \frac {4 i \sqrt {3}}{9}+\frac {37}{72}, \frac {x +i \sqrt {3}}{2 x}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (x +i \sqrt {3}\right )^{-\frac {5 i \sqrt {3}}{36}} \operatorname {HeunC}\left (\frac {10 i \sqrt {3}}{9}, -1-\frac {5 i \sqrt {3}}{18}, 1-\frac {5 i \sqrt {3}}{18}, -\frac {8 i \sqrt {3}}{9}, \frac {4 i \sqrt {3}}{9}+\frac {37}{72}, \frac {1}{2}+\frac {i \sqrt {3}}{2 x}\right ) \left (-x +i \sqrt {3}\right )^{1-\frac {5 i \sqrt {3}}{36}} \left (2 x \right )^{\frac {5 i \sqrt {3}}{18}} {\mathrm e}^{\frac {5 i \sqrt {3}}{18}+\frac {5 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, x}{3}\right )}{18}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x +i \sqrt {3}\right )^{\frac {5 i \sqrt {3}}{36}} \left (-x +i \sqrt {3}\right )^{\frac {5 i \sqrt {3}}{36}} \left (2 x \right )^{-\frac {5 i \sqrt {3}}{18}} {\mathrm e}^{-\frac {5 i \sqrt {3}}{18}} {\mathrm e}^{-\frac {5 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, x}{3}\right )}{18}} y}{\operatorname {HeunC}\left (\frac {10 i \sqrt {3}}{9}, -1-\frac {5 i \sqrt {3}}{18}, 1-\frac {5 i \sqrt {3}}{18}, -\frac {8 i \sqrt {3}}{9}, \frac {4 i \sqrt {3}}{9}+\frac {37}{72}, \frac {x +i \sqrt {3}}{2 x}\right ) \left (-x +i \sqrt {3}\right )}\right ] \\ \end{align*}