Link to actual problem [5557] \[ \boxed {x \left (x +3\right )^{2} y^{\prime \prime }-y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. Difference is 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 &= x \left (1+\frac {x}{3}\right )^{\frac {1}{2}-\frac {i \sqrt {3}}{6}} \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {i \sqrt {3}}{6}, -\frac {i \sqrt {3}}{6}+\frac {3}{2}\right ], \left [-\frac {i \sqrt {3}}{3}+1\right ], 1+\frac {x}{3}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {3 \left (1+\frac {x}{3}\right )^{\frac {i \sqrt {3}}{6}} y}{x \sqrt {3 x +9}\, \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {i \sqrt {3}}{6}, -\frac {i \sqrt {3}}{6}+\frac {3}{2}\right ], \left [-\frac {i \sqrt {3}}{3}+1\right ], 1+\frac {x}{3}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x \left (1+\frac {x}{3}\right )^{\frac {1}{2}+\frac {i \sqrt {3}}{6}} \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {i \sqrt {3}}{6}, \frac {3}{2}+\frac {i \sqrt {3}}{6}\right ], \left [1+\frac {i \sqrt {3}}{3}\right ], 1+\frac {x}{3}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {3 \left (1+\frac {x}{3}\right )^{-\frac {i \sqrt {3}}{6}} y}{x \sqrt {3 x +9}\, \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {i \sqrt {3}}{6}, \frac {3}{2}+\frac {i \sqrt {3}}{6}\right ], \left [1+\frac {i \sqrt {3}}{3}\right ], 1+\frac {x}{3}\right )}\right ] \\ \end{align*}