Link to actual problem [2543] \[ \boxed {\left (-z^{2}+1\right ) y^{\prime \prime }-z y^{\prime }+m^{2} y=0} \] With the expansion point for the power series method at \(z = 0\).
type detected by program
{"second order series method. Ordinary point", "second_order_change_of_variable_on_x_method_1", "second_order_change_of_variable_on_x_method_2", "second order series method. Taylor series method"}
type detected by Maple
[_Gegenbauer, [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
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 &= {\mathrm e}^{\frac {i \sqrt {-\frac {m^{2}}{\left (z -1\right ) \left (z +1\right )}}\, \left (z -1\right ) \left (z +1\right ) \ln \left (z +\sqrt {z^{2}-1}\right )}{\sqrt {\left (z -1\right ) \left (z +1\right )}}}\right ] \\ \left [R &= z, S \left (R \right ) &= \left (z +\sqrt {z^{2}-1}\right )^{-\frac {i \sqrt {-\frac {m^{2}}{\left (z -1\right ) \left (z +1\right )}}\, z^{2}}{\sqrt {z^{2}-1}}} \left (z +\sqrt {z^{2}-1}\right )^{\frac {i \sqrt {-\frac {m^{2}}{\left (z -1\right ) \left (z +1\right )}}}{\sqrt {z^{2}-1}}} y\right ] \\ \end{align*}