Link to actual problem [11096] \[ \boxed {y^{\prime \prime }-y^{\prime } a +b \,{\mathrm e}^{2 a x} y=0} \]
type detected by program
{"second_order_bessel_ode_form_A", "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}^{\frac {i \sqrt {b \,{\mathrm e}^{2 x a}}}{a}}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{-\frac {i \sqrt {b \,{\mathrm e}^{2 x a}}}{a}} y\right ] \\ \end{align*}