Link to actual problem [14885] \[ \boxed {\left (2 x^{2}-1\right ) y^{\prime \prime }+2 y^{\prime } x -3 y=0} \] 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 {3}{2 x^{2}-1}}\, \sqrt {2 x^{2}-1}\, \ln \left (\sqrt {2}\, x +\sqrt {2 x^{2}-1}\right ) \sqrt {2}}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \left (\sqrt {2}\, x +\sqrt {2 x^{2}-1}\right )^{-\frac {i \sqrt {2 x^{2}-1}\, \sqrt {-\frac {6}{2 x^{2}-1}}}{2}} y\right ] \\ \end{align*}