Link to actual problem [1298] \[ \boxed {4 x^{2} y^{\prime \prime }+x \left (4 x^{2}+2 x +7\right ) y^{\prime }-\left (-7 x^{2}-4 x +1\right ) y=0} \] 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 {\operatorname {HeunB}\left (-\frac {5}{4}, \frac {\sqrt {2}}{2}, \frac {3}{4}, -\frac {9 \sqrt {2}}{8}, \frac {\sqrt {2}\, x}{2}\right ) {\mathrm e}^{-\frac {\left (1+x \right ) x}{2}}}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x \,{\mathrm e}^{\frac {\left (1+x \right ) x}{2}} y}{\operatorname {HeunB}\left (-\frac {5}{4}, \frac {\sqrt {2}}{2}, \frac {3}{4}, -\frac {9 \sqrt {2}}{8}, \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 &= x^{\frac {1}{4}} \operatorname {HeunB}\left (\frac {5}{4}, \frac {\sqrt {2}}{2}, \frac {3}{4}, -\frac {9 \sqrt {2}}{8}, \frac {\sqrt {2}\, x}{2}\right ) {\mathrm e}^{-\frac {\left (1+x \right ) x}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {\left (1+x \right ) x}{2}} y}{x^{\frac {1}{4}} \operatorname {HeunB}\left (\frac {5}{4}, \frac {\sqrt {2}}{2}, \frac {3}{4}, -\frac {9 \sqrt {2}}{8}, \frac {\sqrt {2}\, x}{2}\right )}\right ] \\ \end{align*}