Link to actual problem [1333] \[ \boxed {x^{2} \left (x^{2}+2\right ) y^{\prime \prime }+x \left (x^{2}+3\right ) y^{\prime }-y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. 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 {\operatorname {hypergeom}\left (\left [\frac {3}{4}, \frac {3}{4}\right ], \left [\frac {1}{4}\right ], -\frac {x^{2}}{2}\right ) \left (x^{2}+2\right )^{\frac {5}{4}}}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x y}{\operatorname {hypergeom}\left (\left [\frac {3}{4}, \frac {3}{4}\right ], \left [\frac {1}{4}\right ], -\frac {x^{2}}{2}\right ) \left (x^{2}+2\right )^{\frac {5}{4}}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {hypergeom}\left (\left [\frac {3}{2}, \frac {3}{2}\right ], \left [\frac {7}{4}\right ], -\frac {x^{2}}{2}\right ) \left (x^{2}+2\right )^{\frac {1}{8}} \sqrt {x}\, \left (-2 x^{2}-4\right )^{\frac {9}{8}}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {y}{2 \operatorname {hypergeom}\left (\left [\frac {3}{2}, \frac {3}{2}\right ], \left [\frac {7}{4}\right ], -\frac {x^{2}}{2}\right ) \left (x^{2}+2\right )^{\frac {9}{8}} \sqrt {x}\, \left (-2 x^{2}-4\right )^{\frac {1}{8}}}\right ] \\ \end{align*}