Link to actual problem [2393] \[ \boxed {2 x^{2} \left (1-3 x \right ) y^{\prime \prime }+5 y^{\prime } x -2 y=0} \] 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 &= 72-\frac {12 x -1}{x^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{2} y}{72 x^{2}-12 x +1}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\left (3 x -1\right )^{\frac {7}{2}} \left (72 x^{2}-12 x +1\right ) \sqrt {3}\, \arcsin \left (\sqrt {3}\, \sqrt {x}\right )}{3 x^{2} \left (1-3 x \right )^{\frac {7}{2}}}+\frac {\left (-24 x^{2}-10 x +1\right ) \sqrt {3 x -1}}{x^{\frac {3}{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {3 x^{\frac {7}{2}} \sqrt {3 x -1}\, y}{\sqrt {3}\, \arcsin \left (\sqrt {3}\, \sqrt {x}\right ) \left (x^{\frac {3}{2}}-12 x^{\frac {5}{2}}+72 x^{\frac {7}{2}}\right ) \sqrt {1-3 x}-216 x^{5}-18 x^{4}+39 x^{3}-3 x^{2}}\right ] \\ \end{align*}