Link to actual problem [1359] \[ \boxed {x^{2} y^{\prime \prime }+x \left (x^{2}+x +1\right ) y^{\prime }+x \left (2-x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. Repeated root"}
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 &= \operatorname {HeunB}\left (0, \sqrt {2}, -4, -3 \sqrt {2}, \frac {\sqrt {2}\, x}{2}\right ) {\mathrm e}^{-x -\frac {1}{2} x^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{x} {\mathrm e}^{\frac {x^{2}}{2}} y}{\operatorname {HeunB}\left (0, \sqrt {2}, -4, -3 \sqrt {2}, \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 &= \operatorname {HeunB}\left (0, \sqrt {2}, -4, -3 \sqrt {2}, \frac {\sqrt {2}\, x}{2}\right ) {\mathrm e}^{-x -\frac {1}{2} x^{2}} \left (\int \frac {{\mathrm e}^{\frac {1}{2} x^{2}+x}}{\operatorname {HeunB}\left (0, \sqrt {2}, -4, -3 \sqrt {2}, \frac {\sqrt {2}\, x}{2}\right )^{2} x}d x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{x} {\mathrm e}^{\frac {x^{2}}{2}} y}{\operatorname {HeunB}\left (0, \sqrt {2}, -4, -3 \sqrt {2}, \frac {\sqrt {2}\, x}{2}\right ) \left (\int \frac {{\mathrm e}^{\frac {x^{2}}{2}} {\mathrm e}^{x}}{\operatorname {HeunB}\left (0, \sqrt {2}, -4, -3 \sqrt {2}, \frac {\sqrt {2}\, x}{2}\right )^{2} x}d x \right )}\right ] \\ \end{align*}