Link to actual problem [2942] \[ \boxed {x^{2} y^{\prime \prime }+\left (-2 x^{5}+9 x \right ) y^{\prime }+\left (10 x^{4}+5 x^{2}+25\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. Complex roots"}
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 &= x^{-4+3 i} \operatorname {HeunB}\left (3 i, 0, 11, \frac {5 \sqrt {2}}{2}, -\frac {\sqrt {2}\, x^{2}}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{4-3 i} y}{\operatorname {HeunB}\left (3 i, 0, 11, \frac {5 \sqrt {2}}{2}, -\frac {\sqrt {2}\, x^{2}}{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{-4-3 i} \operatorname {HeunB}\left (-3 i, 0, 11, \frac {5 \sqrt {2}}{2}, -\frac {\sqrt {2}\, x^{2}}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{4+3 i} y}{\operatorname {HeunB}\left (-3 i, 0, 11, \frac {5 \sqrt {2}}{2}, -\frac {\sqrt {2}\, x^{2}}{2}\right )}\right ] \\ \end{align*}