Link to actual problem [13980] \[ \boxed {\left (-x^{4}+x^{3}\right ) y^{\prime \prime }+\left (-1+3 x \right ) y^{\prime }+827 y=0} \] With the expansion point for the power series method at \(x = 1\).
type detected by program
{"second order series method. Regular singular point. Difference is 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 {{\mathrm e}^{-\frac {3}{4}+\frac {4 x -1}{2 x^{2}}} \operatorname {HeunB}\left (3, \sqrt {2}, -1, 1656 \sqrt {2}, \frac {\sqrt {2}}{2}-\frac {\sqrt {2}}{2 x}\right ) \left (-1+x \right )^{3}}{x^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {3}{4}} {\mathrm e}^{-\frac {4 x -1}{2 x^{2}}} x^{2} y}{\operatorname {HeunB}\left (3, \sqrt {2}, -1, 1656 \sqrt {2}, \frac {\sqrt {2}\, \left (-1+x \right )}{2 x}\right ) \left (-1+x \right )^{3}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {{\mathrm e}^{-\frac {3}{4}+\frac {4 x -1}{2 x^{2}}} \operatorname {HeunB}\left (3, \sqrt {2}, -1, 1656 \sqrt {2}, \frac {\sqrt {2}}{2}-\frac {\sqrt {2}}{2 x}\right ) \left (\int \frac {x^{2} {\mathrm e}^{\frac {3}{2}-\frac {4 x -1}{2 x^{2}}}}{\operatorname {HeunB}\left (3, \sqrt {2}, -1, 1656 \sqrt {2}, \frac {\sqrt {2}}{2}-\frac {\sqrt {2}}{2 x}\right )^{2} \left (-1+x \right )^{4}}d x \right ) \left (-1+x \right )^{3}}{x^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {3}{4}} {\mathrm e}^{-\frac {4 x -1}{2 x^{2}}} x^{2} y}{\operatorname {HeunB}\left (3, \sqrt {2}, -1, 1656 \sqrt {2}, \frac {\sqrt {2}\, \left (-1+x \right )}{2 x}\right ) \left (\int \frac {x^{2} {\mathrm e}^{\frac {3}{2}} {\mathrm e}^{-\frac {4 x -1}{2 x^{2}}}}{\operatorname {HeunB}\left (3, \sqrt {2}, -1, 1656 \sqrt {2}, \frac {\sqrt {2}\, \left (-1+x \right )}{2 x}\right )^{2} \left (-1+x \right )^{4}}d x \right ) \left (-1+x \right )^{3}}\right ] \\ \end{align*}