Link to actual problem [745] \[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime } x +\alpha ^{2} y=0} \] With the expansion point for the power series method at \(x = 0\).
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 {\alpha ^{2}}{\left (-1+x \right ) \left (1+x \right )}}\, \left (-1+x \right ) \left (1+x \right ) \ln \left (x +\sqrt {x^{2}-1}\right )}{\sqrt {\left (-1+x \right ) \left (1+x \right )}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \left (x +\sqrt {x^{2}-1}\right )^{-\frac {i \sqrt {-\frac {\alpha ^{2}}{\left (-1+x \right ) \left (1+x \right )}}\, x^{2}}{\sqrt {x^{2}-1}}} \left (x +\sqrt {x^{2}-1}\right )^{\frac {i \sqrt {-\frac {\alpha ^{2}}{\left (-1+x \right ) \left (1+x \right )}}}{\sqrt {x^{2}-1}}} y\right ] \\ \end{align*}