Link to actual problem [6931] \[ \boxed {2 x^{2} y^{\prime \prime }-3 x \left (1-x \right ) y^{\prime }+2 y=0} \] 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 &= \sqrt {x}\, {\mathrm e}^{-\frac {3 x}{2}} \left (1+3 x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {3 x}{2}} y}{\sqrt {x}\, \left (1+3 x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (3 x^{\frac {3}{2}} \sqrt {6}+\sqrt {x}\, \sqrt {6}\right ) \sqrt {\pi }\, {\mathrm e}^{-\frac {3 x}{2}} \operatorname {erfi}\left (\frac {\sqrt {x}\, \sqrt {6}}{2}\right )-6 x\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {3 x}{2}} y}{\sqrt {6}\, \sqrt {\pi }\, \left (\sqrt {x}+3 x^{\frac {3}{2}}\right ) \operatorname {erfi}\left (\frac {\sqrt {x}\, \sqrt {6}}{2}\right )-6 x \,{\mathrm e}^{\frac {3 x}{2}}}\right ] \\ \end{align*}