Link to actual problem [1356] \[ \boxed {4 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+12 x^{2} \left (x +1\right ) y^{\prime }+\left (3 x^{2}+3 x +1\right ) y=0} \] 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 {\sqrt {i \sqrt {3}-2 x -1}\, {\mathrm e}^{-\frac {\sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, x}{3}+\frac {\sqrt {3}}{3}\right )}{2}} \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {\sqrt {\frac {\sqrt {3}+i}{i-\sqrt {3}}}}{2}-\sqrt {\frac {\sqrt {3}+i}{\left (-i+\sqrt {3}\right )^{3}}}, \frac {1}{2}+\frac {\sqrt {\frac {\sqrt {3}+i}{i-\sqrt {3}}}}{2}-\sqrt {\frac {\sqrt {3}+i}{\left (-i+\sqrt {3}\right )^{3}}}\right ], \left [1-2 \sqrt {\frac {\sqrt {3}+i}{\left (-i+\sqrt {3}\right )^{3}}}\right ], -\frac {i \sqrt {3}\, x -x -2}{i \sqrt {3}\, x +x +2}\right ) \sqrt {x}\, \left (-\frac {2 i x +i-\sqrt {3}}{\sqrt {3}+2 i x +i}\right )^{\frac {1}{4}-\frac {i \sqrt {3}}{4}}}{\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}} {\mathrm e}^{\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (2 x +1\right )}{3}\right )}{2}} \left (\frac {\sqrt {3}-2 i x -i}{\sqrt {3}+2 i x +i}\right )^{\frac {i \sqrt {3}}{4}} y}{\sqrt {i \sqrt {3}-2 x -1}\, \sqrt {x}\, \left (\frac {\sqrt {3}-2 i x -i}{\sqrt {3}+2 i x +i}\right )^{\frac {1}{4}}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\sqrt {i \sqrt {3}-2 x -1}\, {\mathrm e}^{-\frac {\sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, x}{3}+\frac {\sqrt {3}}{3}\right )}{2}} \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {\sqrt {\frac {\sqrt {3}+i}{i-\sqrt {3}}}}{2}+\sqrt {\frac {\sqrt {3}+i}{\left (-i+\sqrt {3}\right )^{3}}}, \frac {1}{2}+\frac {\sqrt {\frac {\sqrt {3}+i}{i-\sqrt {3}}}}{2}+\sqrt {\frac {\sqrt {3}+i}{\left (-i+\sqrt {3}\right )^{3}}}\right ], \left [1+2 \sqrt {\frac {\sqrt {3}+i}{\left (-i+\sqrt {3}\right )^{3}}}\right ], -\frac {i \sqrt {3}\, x -x -2}{i \sqrt {3}\, x +x +2}\right ) \sqrt {x}\, \left (-\frac {2 i x +i-\sqrt {3}}{\sqrt {3}+2 i x +i}\right )^{\frac {3}{4}+\frac {i \sqrt {3}}{4}}}{\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}} {\mathrm e}^{\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (2 x +1\right )}{3}\right )}{2}} \left (\frac {\sqrt {3}-2 i x -i}{\sqrt {3}+2 i x +i}\right )^{-\frac {i \sqrt {3}}{4}} y}{\sqrt {i \sqrt {3}-2 x -1}\, \operatorname {hypergeom}\left (\left [1, \frac {1}{2}+\frac {i \sqrt {3}}{2}\right ], \left [\frac {i \sqrt {3}}{2}+\frac {3}{2}\right ], \frac {-i \sqrt {3}\, x +x +2}{i \sqrt {3}\, x +x +2}\right ) \sqrt {x}\, \left (\frac {\sqrt {3}-2 i x -i}{\sqrt {3}+2 i x +i}\right )^{\frac {3}{4}}}\right ] \\ \end{align*}