Link to actual problem [1294] \[ \boxed {x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }-x \left (4 x^{2}+3\right ) y^{\prime }+\left (2+2 x \right ) y=0} \] 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 &= \frac {\sqrt {x}\, \operatorname {HeunG}\left (-1, -\sqrt {2}, -3, 0, -\frac {1}{2}, -\frac {3}{4}, -\frac {\sqrt {2}\, x}{2}\right )}{\left (x^{2}-2\right )^{\frac {7}{4}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x^{2}-2\right )^{\frac {7}{4}} y}{\sqrt {x}\, \operatorname {HeunG}\left (-1, -\sqrt {2}, -3, 0, -\frac {1}{2}, -\frac {3}{4}, -\frac {\sqrt {2}\, x}{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {x^{2} \operatorname {HeunG}\left (-1, -\sqrt {2}, -\frac {3}{2}, \frac {3}{2}, \frac {5}{2}, -\frac {3}{4}, -\frac {\sqrt {2}\, x}{2}\right )}{\left (x^{2}-2\right )^{\frac {7}{4}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x^{2}-2\right )^{\frac {7}{4}} y}{x^{2} \operatorname {HeunG}\left (-1, -\sqrt {2}, -\frac {3}{2}, \frac {3}{2}, \frac {5}{2}, -\frac {3}{4}, -\frac {\sqrt {2}\, x}{2}\right )}\right ] \\ \end{align*}