Link to actual problem [11114] \[ \boxed {y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+b \right ) y^{\prime }+\left (\alpha \,{\mathrm e}^{2 \lambda x}+\beta \,{\mathrm e}^{\lambda x}+\gamma \right ) y=0} \]
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 &= \operatorname {WhittakerM}\left (-\frac {a b +a \lambda -2 \beta }{2 \lambda \sqrt {a^{2}-4 \alpha }}, \frac {\sqrt {b^{2}-4 \gamma }}{2 \lambda }, \frac {\sqrt {a^{2}-4 \alpha }\, {\mathrm e}^{\lambda x}}{\lambda }\right ) {\mathrm e}^{-\frac {b \lambda x +x \,\lambda ^{2}+{\mathrm e}^{\lambda x} a}{2 \lambda }}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {{\mathrm e}^{\lambda x} a +x \lambda \left (b +\lambda \right )}{2 \lambda }} y}{\operatorname {WhittakerM}\left (-\frac {a \left (b +\lambda \right )-2 \beta }{2 \sqrt {a^{2}-4 \alpha }\, \lambda }, \frac {\sqrt {b^{2}-4 \gamma }}{2 \lambda }, \frac {\sqrt {a^{2}-4 \alpha }\, {\mathrm e}^{\lambda x}}{\lambda }\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {WhittakerW}\left (-\frac {a b +a \lambda -2 \beta }{2 \lambda \sqrt {a^{2}-4 \alpha }}, \frac {\sqrt {b^{2}-4 \gamma }}{2 \lambda }, \frac {\sqrt {a^{2}-4 \alpha }\, {\mathrm e}^{\lambda x}}{\lambda }\right ) {\mathrm e}^{-\frac {b \lambda x +x \,\lambda ^{2}+{\mathrm e}^{\lambda x} a}{2 \lambda }}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {{\mathrm e}^{\lambda x} a +x \lambda \left (b +\lambda \right )}{2 \lambda }} y}{\operatorname {WhittakerW}\left (-\frac {a \left (b +\lambda \right )-2 \beta }{2 \sqrt {a^{2}-4 \alpha }\, \lambda }, \frac {\sqrt {b^{2}-4 \gamma }}{2 \lambda }, \frac {\sqrt {a^{2}-4 \alpha }\, {\mathrm e}^{\lambda x}}{\lambda }\right )}\right ] \\ \end{align*}