Link to actual problem [6484] \[ \boxed {\left (x^{2}+1\right ) x^{2} y^{\prime \prime }-y^{\prime } x +\left (x +2\right ) y=0} \] 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 &= \frac {x^{1+i} \left (i x +1\right )^{\frac {1}{4}} \operatorname {HeunG}\left (-1, -i, i, 1+i, 1+2 i, \frac {1}{2}, i x \right )}{\left (-i+x \right )^{\frac {1}{4}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{-1-i} \left (-i+x \right )^{\frac {1}{4}} y}{\left (i x +1\right )^{\frac {1}{4}} \operatorname {HeunG}\left (-1, -i, i, 1+i, 1+2 i, \frac {1}{2}, i x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {x^{1-i} \left (i x +1\right )^{\frac {1}{4}} \operatorname {HeunG}\left (-1, -i, -i, 1-i, 1-2 i, \frac {1}{2}, i x \right )}{\left (-i+x \right )^{\frac {1}{4}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{-1+i} \left (-i+x \right )^{\frac {1}{4}} y}{\left (i x +1\right )^{\frac {1}{4}} \operatorname {HeunG}\left (-1, -i, -i, 1-i, 1-2 i, \frac {1}{2}, i x \right )}\right ] \\ \end{align*}