Link to actual problem [6442] \[ \boxed {x^{2} \left (x^{2}-1\right ) y^{\prime \prime }-x \left (1-x \right ) y^{\prime }+2 y=0} \] 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 &= x^{\sqrt {2}} \operatorname {HeunG}\left (-1, -\sqrt {2}, \sqrt {2}, \sqrt {2}-1, 1+2 \sqrt {2}, 0, x\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{-\sqrt {2}} y}{\operatorname {HeunG}\left (-1, -\sqrt {2}, \sqrt {2}, \sqrt {2}-1, 1+2 \sqrt {2}, 0, x\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{-\sqrt {2}} \operatorname {HeunG}\left (-1, \sqrt {2}, -\sqrt {2}, -1-\sqrt {2}, 1-2 \sqrt {2}, 0, x\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\sqrt {2}} y}{\operatorname {HeunG}\left (-1, \sqrt {2}, -\sqrt {2}, -1-\sqrt {2}, 1-2 \sqrt {2}, 0, x\right )}\right ] \\ \end{align*}