Link to actual problem [6932] \[ \boxed {2 x^{2} y^{\prime \prime }+x \left (4 x -1\right ) y^{\prime }+2 \left (3 x -1\right ) y=0} \] 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 [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{2 x} y}{x^{2}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= -\frac {-16 \,\operatorname {erfi}\left (\sqrt {x}\, \sqrt {2}\right ) \sqrt {2}\, \sqrt {\pi }\, x^{\frac {5}{2}} {\mathrm e}^{-2 x}+16 x^{2}+4 x +3}{\sqrt {x}}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {\sqrt {x}\, {\mathrm e}^{2 x} y}{16 \left (-\sqrt {\pi }\, x^{\frac {5}{2}} \operatorname {erfi}\left (\sqrt {x}\, \sqrt {2}\right ) \sqrt {2}+{\mathrm e}^{2 x} \left (x^{2}+\frac {1}{4} x +\frac {3}{16}\right )\right )}\right ] \\ \end{align*}