Link to actual problem [12407] \[ \boxed {2 x^{2} y^{\prime \prime }+x \left (1-x \right ) y^{\prime }-y=0} \] 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 {{\mathrm e}^{\frac {x}{2}}}{\sqrt {x}}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{-\frac {x}{2}} \sqrt {x}\, y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \sqrt {2}-\frac {\sqrt {\pi }\, {\mathrm e}^{\frac {x}{2}} \left (-1+\operatorname {erf}\left (\frac {\sqrt {x}\, \sqrt {2}}{2}\right )\right )}{\sqrt {x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {x}\, y}{-\sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {x}\, \sqrt {2}}{2}\right ) {\mathrm e}^{\frac {x}{2}}+\sqrt {x}\, \sqrt {2}+\sqrt {\pi }\, {\mathrm e}^{\frac {x}{2}}}\right ] \\ \end{align*}