Link to actual problem [10997] \[ \boxed {\left (2 a x +x^{2}+b \right ) y^{\prime \prime }+\left (x +a \right ) y^{\prime }-y m^{2}=0} \]
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 {-\frac {m^{2}}{2 x a +x^{2}+b}}\, \sqrt {2 x a +x^{2}+b}\, \ln \left (a +x +\sqrt {2 x a +x^{2}+b}\right )}\right ] \\ \left [R &= x, S \left (R \right ) &= \left (a +x +\sqrt {2 x a +x^{2}+b}\right )^{-i \sqrt {-\frac {m^{2}}{2 x a +x^{2}+b}}\, \sqrt {2 x a +x^{2}+b}} y\right ] \\ \end{align*}