Link to actual problem [2396] \[ \boxed {\left (8-x \right ) x^{2} y^{\prime \prime }+6 y^{\prime } x -y=0} \] 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 {\left (-8+x \right )^{\frac {7}{4}} \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {3}{2}\right ], \left [\frac {1}{4}\right ], \frac {x}{8}\right )}{x^{\frac {1}{4}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {1}{4}} y}{\left (-8+x \right )^{\frac {7}{4}} \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {3}{2}\right ], \left [\frac {1}{4}\right ], \frac {x}{8}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \sqrt {x}\, \left (-8+x \right )^{\frac {7}{4}} \operatorname {hypergeom}\left (\left [\frac {5}{4}, \frac {9}{4}\right ], \left [\frac {7}{4}\right ], \frac {x}{8}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\sqrt {x}\, \left (-8+x \right )^{\frac {7}{4}} \operatorname {hypergeom}\left (\left [\frac {5}{4}, \frac {9}{4}\right ], \left [\frac {7}{4}\right ], \frac {x}{8}\right )}\right ] \\ \end{align*}