Link to actual problem [11106] \[ \boxed {y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}-\lambda \right ) y^{\prime }+y b \,{\mathrm e}^{2 \lambda x}=0} \]
type detected by program
{"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*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{\frac {\left (-a +\sqrt {a^{2}-4 b}\right ) {\mathrm e}^{\lambda x}}{2 \lambda }}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{-\frac {\left (-a +\sqrt {a^{2}-4 b}\right ) {\mathrm e}^{\lambda x}}{2 \lambda }} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {\left (a +\sqrt {a^{2}-4 b}\right ) {\mathrm e}^{\lambda x}}{2 \lambda }}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{\frac {\left (a +\sqrt {a^{2}-4 b}\right ) {\mathrm e}^{\lambda x}}{2 \lambda }} y\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= y, S \left (R \right ) &= \frac {{\mathrm e}^{\lambda x}}{\lambda }\right ] \\ \end{align*}