Link to actual problem [2395] \[ \boxed {\left (4+x \right ) x^{2} y^{\prime \prime }+x \left (x -1\right ) y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. Difference 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 {x^{\frac {1}{4}}}{\left (x +4\right )^{\frac {1}{4}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x +4\right )^{\frac {1}{4}} y}{x^{\frac {1}{4}}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {hypergeom}\left (\left [1, 1\right ], \left [\frac {7}{4}\right ], -\frac {x}{4}\right ) x\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {hypergeom}\left (\left [1, 1\right ], \left [\frac {7}{4}\right ], -\frac {x}{4}\right ) x}\right ] \\ \end{align*}