Link to actual problem [4707] \[ \boxed {x^{2} \left (1-4 x \right ) y^{\prime \prime }+\left (\left (1-n \right ) x -\left (6-4 n \right ) x^{2}\right ) y^{\prime }+n \left (1-n \right ) x y=0} \] 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 &= \left (\frac {\left (-\sqrt {4 x -1}+i\right ) x}{\sqrt {4 x -1}+i}\right )^{\frac {n}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \left (\frac {1}{2}+\frac {i \sqrt {4 x -1}}{2}-x \right )^{-\frac {n}{2}} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (\frac {\left (\sqrt {4 x -1}+i\right ) x}{-\sqrt {4 x -1}+i}\right )^{\frac {n}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \left (\frac {1}{2}-\frac {i \sqrt {4 x -1}}{2}-x \right )^{-\frac {n}{2}} y\right ] \\ \end{align*}