Link to actual problem [11105] \[ \boxed {y^{\prime \prime }+\left (a \,{\mathrm e}^{2 \lambda x}+\lambda \right ) y^{\prime }-a \lambda \,{\mathrm e}^{2 \lambda x} y=0} \]
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 [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\lambda x} y}{a \,{\mathrm e}^{2 \lambda x}+\lambda }\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \sqrt {\lambda }\, \sqrt {2}\, \sqrt {a}\, {\mathrm e}^{-\frac {a \,{\mathrm e}^{2 \lambda x}}{2 \lambda }}+a \,\operatorname {erf}\left (\frac {\sqrt {2}\, {\mathrm e}^{\lambda x} \sqrt {a}}{2 \sqrt {\lambda }}\right ) \sqrt {\pi }\, {\mathrm e}^{\lambda x}+\sqrt {\pi }\, \lambda \,\operatorname {erf}\left (\frac {\sqrt {2}\, {\mathrm e}^{\lambda x} \sqrt {a}}{2 \sqrt {\lambda }}\right ) {\mathrm e}^{-\lambda x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {a \,{\mathrm e}^{2 \lambda x}}{2 \lambda }} {\mathrm e}^{\lambda x} y}{{\mathrm e}^{\frac {a \,{\mathrm e}^{2 \lambda x}}{2 \lambda }} \sqrt {\pi }\, \left (a \,{\mathrm e}^{2 \lambda x}+\lambda \right ) \operatorname {erf}\left (\frac {\sqrt {2}\, {\mathrm e}^{\lambda x} \sqrt {a}}{2 \sqrt {\lambda }}\right )+\sqrt {a}\, \sqrt {\lambda }\, \sqrt {2}\, {\mathrm e}^{\lambda x}}\right ] \\ \end{align*}