Link to actual problem [7235] \[ \boxed {\left (x -2\right ) y^{\prime \prime }+\frac {y^{\prime }}{x}+\left (1+x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. 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 &= {\mathrm e}^{i x} \operatorname {HeunC}\left (4 i, -\frac {3}{2}, -\frac {1}{2}, 6, \frac {5}{8}, \frac {x}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-i x} y}{\operatorname {HeunC}\left (4 i, -\frac {3}{2}, -\frac {1}{2}, 6, \frac {5}{8}, \frac {x}{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{i x} \operatorname {HeunC}\left (4 i, \frac {3}{2}, -\frac {1}{2}, 6, \frac {5}{8}, \frac {x}{2}\right ) x^{\frac {3}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-i x} y}{\operatorname {HeunC}\left (4 i, \frac {3}{2}, -\frac {1}{2}, 6, \frac {5}{8}, \frac {x}{2}\right ) x^{\frac {3}{2}}}\right ] \\ \end{align*}