Link to actual problem [11075] \[ \boxed {x \left (x^{2 n}+a \right ) y^{\prime \prime }+\left (x^{2 n}+a -n a \right ) y^{\prime }-b^{2} x^{2 n -1} y=0} \]
type detected by program
{"second_order_change_of_variable_on_x_method_1"}
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 \left (\int \sqrt {-\frac {b^{2} x^{2 n -2}}{x^{2 n}+a}}d x \right )}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{\int -i \sqrt {-\frac {b^{2} x^{2 n -2}}{x^{2 n}+a}}d x} y\right ] \\ \end{align*}