Link to actual problem [9551] \[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+y^{\prime } x +2 y=0} \]
type detected by program
{"kovacic", "second_order_change_of_variable_on_x_method_1", "second_order_change_of_variable_on_x_method_2"}
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 {2}\, \sqrt {\frac {1}{x^{2}+1}}\, \sqrt {x^{2}+1}\, \operatorname {arcsinh}\left (x \right )}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{-i \sqrt {2}\, \sqrt {\frac {1}{x^{2}+1}}\, \sqrt {x^{2}+1}\, \operatorname {arcsinh}\left (x \right )} y\right ] \\ \end{align*}