Link to actual problem [6045] \[ \boxed {\left (x^{2}+x -2\right )^{2} y^{\prime \prime }+3 \left (x +2\right ) y^{\prime }+\left (x -1\right ) y=0} \] With the expansion point for the power series method at \(x = -2\).
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 {\left (2+x \right ) \operatorname {HeunC}\left (-\frac {1}{3}, \frac {4}{3}, 1, \frac {7}{18}, \frac {4}{9}, \frac {2+x}{-1+x}\right )}{-1+x}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {\left (-1+x \right ) y}{\left (2+x \right ) \operatorname {HeunC}\left (-\frac {1}{3}, \frac {4}{3}, 1, \frac {7}{18}, \frac {4}{9}, \frac {2+x}{-1+x}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\left (\left (-1+x \right )^{2}\right )^{\frac {1}{6}} \operatorname {HeunC}\left (-\frac {1}{3}, -\frac {4}{3}, 1, \frac {7}{18}, \frac {4}{9}, \frac {2+x}{-1+x}\right )}{\left (2+x \right )^{\frac {1}{3}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (2+x \right )^{\frac {1}{3}} y}{\left (\left (-1+x \right )^{2}\right )^{\frac {1}{6}} \operatorname {HeunC}\left (-\frac {1}{3}, -\frac {4}{3}, 1, \frac {7}{18}, \frac {4}{9}, \frac {2+x}{-1+x}\right )}\right ] \\ \end{align*}