Link to actual problem [11100] \[ \boxed {y^{\prime \prime }+2 a \,{\mathrm e}^{\lambda x} y^{\prime }+a \,{\mathrm e}^{\lambda x} \left (a \,{\mathrm e}^{\lambda x}+\lambda \right ) y=0} \]
type detected by program
{"kovacic", "second_order_change_of_variable_on_y_method_1", "linear_second_order_ode_solved_by_an_integrating_factor"}
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*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {a \,{\mathrm e}^{\lambda x}}{\lambda }}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{\frac {a \,{\mathrm e}^{\lambda x}}{\lambda }} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {a \,{\mathrm e}^{\lambda x}}{\lambda }} x\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {a \,{\mathrm e}^{\lambda x}}{\lambda }} y}{x}\right ] \\ \end{align*}