Link to actual problem [1420] \[ \boxed {x \left (x +1\right ) y^{\prime \prime }+y^{\prime } x +y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Regular singular point. Difference is 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 &= x^{i} \operatorname {hypergeom}\left (\left [-i, 1-i\right ], \left [1-2 i\right ], -\frac {1}{x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{-i} y}{\operatorname {hypergeom}\left (\left [-i, 1-i\right ], \left [1-2 i\right ], -\frac {1}{x}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{-i} \operatorname {hypergeom}\left (\left [i, 1+i\right ], \left [1+2 i\right ], -\frac {1}{x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{i} y}{\operatorname {hypergeom}\left (\left [i, 1+i\right ], \left [1+2 i\right ], -\frac {1}{x}\right )}\right ] \\ \end{align*}