Link to actual problem [1357] \[ \boxed {x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }-x \left (-2 x^{2}-4 x +1\right ) y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. Repeated root"}
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 {\left (i \sqrt {3}-2 x -1\right )^{\frac {1}{4}+\frac {7 i \sqrt {3}}{12}} \left (i \sqrt {3}+2 x +1\right )^{\frac {1}{4}-\frac {7 i \sqrt {3}}{12}} x \,{\mathrm e}^{-\frac {7 \sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, x}{3}+\frac {\sqrt {3}}{3}\right )}{6}}}{\left (x^{2}+x +1\right )^{\frac {3}{4}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x^{2}+x +1\right )^{\frac {3}{4}} \left (i \sqrt {3}-2 x -1\right )^{-\frac {7 i \sqrt {3}}{12}} \left (i \sqrt {3}+2 x +1\right )^{\frac {7 i \sqrt {3}}{12}} {\mathrm e}^{\frac {7 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (2 x +1\right )}{3}\right )}{6}} y}{\left (i \sqrt {3}-2 x -1\right )^{\frac {1}{4}} \left (i \sqrt {3}+2 x +1\right )^{\frac {1}{4}} x}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\left (i \sqrt {3}-2 x -1\right )^{-\frac {1}{4}-\frac {7 i \sqrt {3}}{12}} \left (i \sqrt {3}+2 x +1\right )^{\frac {3}{4}+\frac {7 i \sqrt {3}}{12}} x \,{\mathrm e}^{-\frac {7 \sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, x}{3}+\frac {\sqrt {3}}{3}\right )}{6}} \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {\sqrt {3}\, \sqrt {\frac {i-15 \sqrt {3}}{-i+\sqrt {3}}}}{6}+\frac {\sqrt {3}\, \sqrt {\frac {7 \sqrt {3}+23 i}{\left (-i+\sqrt {3}\right )^{3}}}}{3}, \frac {1}{2}+\frac {\sqrt {3}\, \sqrt {\frac {i-15 \sqrt {3}}{-i+\sqrt {3}}}}{6}+\frac {\sqrt {3}\, \sqrt {\frac {7 \sqrt {3}+23 i}{\left (-i+\sqrt {3}\right )^{3}}}}{3}\right ], \left [1+\frac {2 \sqrt {3}\, \sqrt {\frac {7 \sqrt {3}+23 i}{\left (-i+\sqrt {3}\right )^{3}}}}{3}\right ], -\frac {i \sqrt {3}\, x -x -2}{i \sqrt {3}\, x +x +2}\right )}{\left (x^{2}+x +1\right )^{\frac {3}{4}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x^{2}+x +1\right )^{\frac {3}{4}} \left (i \sqrt {3}-2 x -1\right )^{\frac {1}{4}} \left (i \sqrt {3}-2 x -1\right )^{\frac {7 i \sqrt {3}}{12}} \left (i \sqrt {3}+2 x +1\right )^{-\frac {7 i \sqrt {3}}{12}} {\mathrm e}^{\frac {7 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (2 x +1\right )}{3}\right )}{6}} y}{\left (i \sqrt {3}+2 x +1\right )^{\frac {3}{4}} x \operatorname {hypergeom}\left (\left [1, \frac {1}{2}+\frac {7 i \sqrt {3}}{6}\right ], \left [\frac {3}{2}+\frac {7 i \sqrt {3}}{6}\right ], \frac {-i \sqrt {3}\, x +x +2}{i \sqrt {3}\, x +x +2}\right )}\right ] \\ \end{align*}