Link to actual problem [11111] \[ \boxed {y^{\prime \prime }+\left (2 a \,{\mathrm e}^{\lambda x}+b \right ) y^{\prime }+\left (a^{2} {\mathrm e}^{2 \lambda x}+a \left (b +\lambda \right ) {\mathrm e}^{\lambda x}+c \right ) y=0} \]
type detected by program
{"second_order_change_of_variable_on_y_method_1"}
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 {b x}{2}-\frac {a \,{\mathrm e}^{\lambda x}}{\lambda }} \sinh \left (\frac {\sqrt {b^{2}-4 c}\, x}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {b x}{2}} {\mathrm e}^{\frac {a \,{\mathrm e}^{\lambda x}}{\lambda }} y}{\sinh \left (\frac {\sqrt {b^{2}-4 c}\, x}{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {b x}{2}-\frac {a \,{\mathrm e}^{\lambda x}}{\lambda }} \cosh \left (\frac {\sqrt {b^{2}-4 c}\, x}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {b x}{2}} {\mathrm e}^{\frac {a \,{\mathrm e}^{\lambda x}}{\lambda }} y}{\cosh \left (\frac {\sqrt {b^{2}-4 c}\, x}{2}\right )}\right ] \\ \end{align*}