Link to actual problem [6444] \[ \boxed {\left (3 x +1\right ) x y^{\prime \prime }-\left (1+x \right ) y^{\prime }+2 y=0} \] 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^{2} \operatorname {hypergeom}\left (\left [\frac {4}{3}+\frac {i \sqrt {2}}{3}, \frac {4}{3}-\frac {i \sqrt {2}}{3}\right ], \left [\frac {2}{3}\right ], 1+3 x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x^{2} \operatorname {hypergeom}\left (\left [\frac {4}{3}+\frac {i \sqrt {2}}{3}, \frac {4}{3}-\frac {i \sqrt {2}}{3}\right ], \left [\frac {2}{3}\right ], 1+3 x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (1+3 x \right )^{\frac {1}{3}} x^{2} \operatorname {hypergeom}\left (\left [\frac {5}{3}+\frac {i \sqrt {2}}{3}, \frac {5}{3}-\frac {i \sqrt {2}}{3}\right ], \left [\frac {4}{3}\right ], 1+3 x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\left (1+3 x \right )^{\frac {1}{3}} x^{2} \operatorname {hypergeom}\left (\left [\frac {5}{3}+\frac {i \sqrt {2}}{3}, \frac {5}{3}-\frac {i \sqrt {2}}{3}\right ], \left [\frac {4}{3}\right ], 1+3 x \right )}\right ] \\ \end{align*}