Link to actual problem [2382] \[ \boxed {x^{2} \left (4+x \right ) y^{\prime \prime }+7 y^{\prime } x -y=0} \] 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 &= 2+\frac {1}{x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x y}{2 x +1}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {hypergeom}\left (\left [2, 3\right ], \left [\frac {9}{4}\right ], -\frac {x}{4}\right ) x^{\frac {1}{4}} \left (x +4\right )^{\frac {11}{4}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {hypergeom}\left (\left [2, 3\right ], \left [\frac {9}{4}\right ], -\frac {x}{4}\right ) x^{\frac {1}{4}} \left (x +4\right )^{\frac {11}{4}}}\right ] \\ \end{align*}