Link to actual problem [2397] \[ \boxed {2 x^{2} y^{\prime \prime }+x \left (x^{2}+1\right ) y^{\prime }-y \left (x +1\right )=0} \] 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 &= {\mathrm e}^{-\frac {x^{2}}{4}} \operatorname {HeunB}\left (\frac {3}{2}, 0, -\frac {3}{2}, 2, \frac {x}{2}\right ) x\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2}}{4}} y}{\operatorname {HeunB}\left (\frac {3}{2}, 0, -\frac {3}{2}, 2, \frac {x}{2}\right ) x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {{\mathrm e}^{-\frac {x^{2}}{4}} \operatorname {HeunB}\left (-\frac {3}{2}, 0, -\frac {3}{2}, 2, \frac {x}{2}\right )}{\sqrt {x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2}}{4}} \sqrt {x}\, y}{\operatorname {HeunB}\left (-\frac {3}{2}, 0, -\frac {3}{2}, 2, \frac {x}{2}\right )}\right ] \\ \end{align*}