Link to actual problem [2381] \[ \boxed {3 x \left (2+3 x \right ) y^{\prime \prime }-4 y^{\prime }+4 y=0} \] 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 &= \operatorname {hypergeom}\left (\left [-\frac {1}{2}-\frac {i \sqrt {7}}{6}, -\frac {1}{2}+\frac {i \sqrt {7}}{6}\right ], \left [\frac {2}{3}\right ], \frac {3 x}{2}+1\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {hypergeom}\left (\left [-\frac {1}{2}-\frac {i \sqrt {7}}{6}, -\frac {1}{2}+\frac {i \sqrt {7}}{6}\right ], \left [\frac {2}{3}\right ], \frac {3 x}{2}+1\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (12 x +8\right )^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [-\frac {1}{6}-\frac {i \sqrt {7}}{6}, -\frac {1}{6}+\frac {i \sqrt {7}}{6}\right ], \left [\frac {4}{3}\right ], \frac {3 x}{2}+1\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\left (12 x +8\right )^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [-\frac {1}{6}-\frac {i \sqrt {7}}{6}, -\frac {1}{6}+\frac {i \sqrt {7}}{6}\right ], \left [\frac {4}{3}\right ], \frac {3 x}{2}+1\right )}\right ] \\ \end{align*}