Link to actual problem [15415] \[ \boxed {y^{\prime \prime }-y^{\prime }+{\mathrm e}^{2 x} y=x \,{\mathrm e}^{2 x}-1} \] Given that one solution of the ode is \begin {align*} y_1 &= \sin \left ({\mathrm e}^{x}\right ) \end {align*}
type detected by program
{"reduction_of_order", "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]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{i \sqrt {{\mathrm e}^{2 x}}}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{-i \sqrt {{\mathrm e}^{2 x}}} y\right ] \\ \end{align*}
\begin{align*} \\ \\ \end{align*}