Link to actual problem [11102] \[ \boxed {y^{\prime \prime }+a \,{\mathrm e}^{\lambda x} y^{\prime }-b \,{\mathrm e}^{\mu x} \left (a \,{\mathrm e}^{\lambda x}+{\mathrm e}^{\mu x} b +\mu \right ) y=0} \]
type detected by program
{"unknown"}
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 \,{\mathrm e}^{\mu x}}{\mu }}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{-\frac {b \,{\mathrm e}^{\mu x}}{\mu }} y\right ] \\ \end{align*}