Link to actual problem [1304] \[ \boxed {10 x^{2} \left (2 x^{2}+x +1\right ) y^{\prime \prime }+x \left (66 x^{2}+13 x +13\right ) y^{\prime }-\left (10 x^{2}+4 x +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. 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 {{\mathrm e}^{\frac {\sqrt {7}\, \arctan \left (\frac {4 \sqrt {7}\, x}{7}+\frac {\sqrt {7}}{7}\right )}{7}} \operatorname {HeunG}\left (\frac {\sqrt {7}+i}{i-\sqrt {7}}, \frac {-1197 i-1671 \sqrt {7}}{790 \sqrt {7}+21490 i}, 0, \frac {27}{10}, \frac {17}{10}, 1-\frac {i \sqrt {7}}{7}, -\frac {4 x}{i \sqrt {7}+1}\right ) x^{{1}/{5}} \left (i \sqrt {7}+4 x +1\right )^{\frac {1}{2}-\frac {i \sqrt {7}}{14}} \left (i \sqrt {7}-4 x -1\right )^{\frac {1}{2}+\frac {i \sqrt {7}}{14}}}{\sqrt {2 x^{2}+x +1}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {\sqrt {7}\, \arctan \left (\frac {\left (1+4 x \right ) \sqrt {7}}{7}\right )}{7}} \sqrt {2 x^{2}+x +1}\, \left (i \sqrt {7}+4 x +1\right )^{\frac {i \sqrt {7}}{14}} \left (i \sqrt {7}-4 x -1\right )^{-\frac {i \sqrt {7}}{14}} y}{\operatorname {HeunG}\left (\frac {\sqrt {7}+i}{i-\sqrt {7}}, \frac {-1197 i-1671 \sqrt {7}}{790 \sqrt {7}+21490 i}, 0, \frac {27}{10}, \frac {17}{10}, 1-\frac {i \sqrt {7}}{7}, -\frac {4 x}{i \sqrt {7}+1}\right ) x^{{1}/{5}} \sqrt {i \sqrt {7}+4 x +1}\, \sqrt {i \sqrt {7}-4 x -1}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\left (i \sqrt {7}+4 x +1\right )^{\frac {1}{2}-\frac {i \sqrt {7}}{14}} \left (i \sqrt {7}-4 x -1\right )^{\frac {1}{2}+\frac {i \sqrt {7}}{14}} {\mathrm e}^{\frac {\sqrt {7}\, \arctan \left (\frac {4 \sqrt {7}\, x}{7}+\frac {\sqrt {7}}{7}\right )}{7}} \operatorname {HeunG}\left (\frac {\sqrt {7}+i}{i-\sqrt {7}}, \frac {477 i \sqrt {7}-1911}{2390 i \sqrt {7}+8750}, -\frac {7}{10}, 2, \frac {3}{10}, 1-\frac {i \sqrt {7}}{7}, -\frac {4 x}{i \sqrt {7}+1}\right )}{\sqrt {x \left (2 x^{2}+x +1\right )}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (i \sqrt {7}+4 x +1\right )^{\frac {i \sqrt {7}}{14}} \left (i \sqrt {7}-4 x -1\right )^{-\frac {i \sqrt {7}}{14}} \sqrt {2 x^{3}+x^{2}+x}\, {\mathrm e}^{-\frac {\sqrt {7}\, \arctan \left (\frac {\left (1+4 x \right ) \sqrt {7}}{7}\right )}{7}} y}{\sqrt {i \sqrt {7}+4 x +1}\, \sqrt {i \sqrt {7}-4 x -1}\, \operatorname {HeunG}\left (\frac {\sqrt {7}+i}{i-\sqrt {7}}, \frac {477 i \sqrt {7}-1911}{2390 i \sqrt {7}+8750}, -\frac {7}{10}, 2, \frac {3}{10}, 1-\frac {i \sqrt {7}}{7}, -\frac {4 x}{i \sqrt {7}+1}\right )}\right ] \\ \end{align*}