Link to actual problem [1455] \[ \boxed {x^{2} \left (1+2 x^{2}\right ) y^{\prime \prime }-x \left (x^{2}+3\right ) y^{\prime }-2 y x=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. Difference is 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 {\operatorname {HeunG}\left (2, 1-i \sqrt {2}, -2, -\frac {1}{2}, \frac {3}{4}, -3, 1-i \sqrt {2}\, x \right )}{\left (4 x^{2}+2\right )^{\frac {1}{4}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (4 x^{2}+2\right )^{\frac {1}{4}} y}{\operatorname {HeunG}\left (2, 1-i \sqrt {2}, -2, -\frac {1}{2}, \frac {3}{4}, -3, 1-i \sqrt {2}\, x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\left (2 x +i \sqrt {2}\right )^{\frac {1}{4}} \operatorname {HeunG}\left (2, -i \sqrt {2}-\frac {5}{16}, -\frac {7}{4}, -\frac {1}{4}, \frac {5}{4}, -3, 1-i \sqrt {2}\, x \right )}{\left (4 x^{2}+2\right )^{\frac {1}{4}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (4 x^{2}+2\right )^{\frac {1}{4}} y}{\left (2 x +i \sqrt {2}\right )^{\frac {1}{4}} \operatorname {HeunG}\left (2, -i \sqrt {2}-\frac {5}{16}, -\frac {7}{4}, -\frac {1}{4}, \frac {5}{4}, -3, 1-i \sqrt {2}\, x \right )}\right ] \\ \end{align*}