Link to actual problem [11112] \[ \boxed {y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+2 b -\lambda \right ) y^{\prime }+\left ({\mathrm e}^{2 \lambda x} c +a b \,{\mathrm e}^{\lambda x}+b^{2}-\lambda b \right ) y=0} \]
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 {-2 b \lambda x +{\mathrm e}^{\lambda x} \sqrt {a^{2}-4 c}-{\mathrm e}^{\lambda x} a}{2 \lambda }}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{-\frac {-2 b \lambda x +{\mathrm e}^{\lambda x} \sqrt {a^{2}-4 c}-{\mathrm e}^{\lambda x} a}{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 {2 b \lambda x +{\mathrm e}^{\lambda x} \sqrt {a^{2}-4 c}+{\mathrm e}^{\lambda x} a}{2 \lambda }}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{\frac {2 b \lambda x +{\mathrm e}^{\lambda x} \sqrt {a^{2}-4 c}+{\mathrm e}^{\lambda x} a}{2 \lambda }} y\right ] \\ \end{align*}