Link to actual problem [6497] \[ \boxed {9 \left (x -2\right )^{2} \left (x -3\right ) y^{\prime \prime }+6 x \left (x -2\right ) y^{\prime }+16 y=0} \] With the expansion point for the power series method at \(x = \infty \).
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 &= \left (-2+x \right )^{\frac {7}{6}-\frac {\sqrt {113}}{6}} \operatorname {hypergeom}\left (\left [\frac {7}{6}-\frac {\sqrt {113}}{6}, \frac {5}{6}-\frac {\sqrt {113}}{6}\right ], \left [1-\frac {\sqrt {113}}{3}\right ], -2+x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (-2+x \right )^{\frac {\sqrt {113}}{6}} y}{\left (-2+x \right )^{\frac {7}{6}} \operatorname {hypergeom}\left (\left [\frac {7}{6}-\frac {\sqrt {113}}{6}, \frac {5}{6}-\frac {\sqrt {113}}{6}\right ], \left [1-\frac {\sqrt {113}}{3}\right ], -2+x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (-2+x \right )^{\frac {7}{6}+\frac {\sqrt {113}}{6}} \operatorname {hypergeom}\left (\left [\frac {7}{6}+\frac {\sqrt {113}}{6}, \frac {5}{6}+\frac {\sqrt {113}}{6}\right ], \left [1+\frac {\sqrt {113}}{3}\right ], -2+x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (-2+x \right )^{-\frac {\sqrt {113}}{6}} y}{\left (-2+x \right )^{\frac {7}{6}} \operatorname {hypergeom}\left (\left [\frac {7}{6}+\frac {\sqrt {113}}{6}, \frac {5}{6}+\frac {\sqrt {113}}{6}\right ], \left [1+\frac {\sqrt {113}}{3}\right ], -2+x \right )}\right ] \\ \end{align*}