Link to actual problem [6922] \[ \boxed {2 x y^{\prime \prime }+5 \left (1-2 x \right ) y^{\prime }-5 y=0} \] 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 {1+10 x}{x^{\frac {3}{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {3}{2}} y}{1+10 x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (-\frac {10 \sqrt {5}}{\sqrt {x}}-\frac {\sqrt {5}}{x^{\frac {3}{2}}}\right ) \sqrt {\pi }\, \operatorname {erfi}\left (\sqrt {5}\, \sqrt {x}\right )+\frac {10 \,{\mathrm e}^{5 x}}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {x^{\frac {5}{2}} y}{10 \left (\sqrt {5}\, \sqrt {\pi }\, x \left (\frac {1}{10}+x \right ) \operatorname {erfi}\left (\sqrt {5}\, \sqrt {x}\right )-{\mathrm e}^{5 x} x^{\frac {3}{2}}\right )}\right ] \\ \end{align*}