Link to actual problem [4709] \[ \boxed {\left (a^{2}+x^{2}\right ) y^{\prime \prime }+x y^{\prime }-n^{2} y=0} \] 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
[[_2nd_order, _with_linear_symmetries], [_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}^{i \sqrt {-\frac {n^{2}}{a^{2}+x^{2}}}\, \sqrt {a^{2}+x^{2}}\, \ln \left (x +\sqrt {a^{2}+x^{2}}\right )}\right ] \\ \left [R &= x, S \left (R \right ) &= \left (x +\sqrt {a^{2}+x^{2}}\right )^{-i \sqrt {-\frac {n^{2}}{a^{2}+x^{2}}}\, \sqrt {a^{2}+x^{2}}} y\right ] \\ \end{align*}