Link to actual problem [11107] \[ \boxed {y^{\prime \prime }+\left (a \,{\mathrm e}^{\lambda x}+b \right ) y^{\prime }+c \left (a \,{\mathrm e}^{\lambda x}+b -c \right ) y=0} \]
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*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (-\lambda -2 c +b \right )^{2} \operatorname {WhittakerM}\left (-\frac {-\lambda -2 c +b}{2 \lambda }, -\frac {-2 \lambda -2 c +b}{2 \lambda }, \frac {a \,{\mathrm e}^{\lambda x}}{\lambda }\right ) {\mathrm e}^{\frac {-{\mathrm e}^{\lambda x} a -\left (b +3 \lambda \right ) x \lambda }{2 \lambda }}+\lambda \operatorname {WhittakerM}\left (-\frac {b -2 c +\lambda }{2 \lambda }, -\frac {-2 \lambda -2 c +b}{2 \lambda }, \frac {a \,{\mathrm e}^{\lambda x}}{\lambda }\right ) \left (\left (\lambda +2 c -b \right ) {\mathrm e}^{\frac {-{\mathrm e}^{\lambda x} a -\left (b +3 \lambda \right ) x \lambda }{2 \lambda }}+a \,{\mathrm e}^{\frac {-{\mathrm e}^{\lambda x} a -x \lambda \left (b +\lambda \right )}{2 \lambda }}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\left (-\lambda -2 c +b \right )^{2} \operatorname {WhittakerM}\left (-\frac {-\lambda -2 c +b}{2 \lambda }, -\frac {-2 \lambda -2 c +b}{2 \lambda }, \frac {a \,{\mathrm e}^{\lambda x}}{\lambda }\right ) {\mathrm e}^{\frac {-{\mathrm e}^{\lambda x} a -\left (b +3 \lambda \right ) x \lambda }{2 \lambda }}+\lambda \operatorname {WhittakerM}\left (-\frac {b -2 c +\lambda }{2 \lambda }, -\frac {-2 \lambda -2 c +b}{2 \lambda }, \frac {a \,{\mathrm e}^{\lambda x}}{\lambda }\right ) \left (\left (\lambda +2 c -b \right ) {\mathrm e}^{\frac {-{\mathrm e}^{\lambda x} a -\left (b +3 \lambda \right ) x \lambda }{2 \lambda }}+a \,{\mathrm e}^{\frac {-{\mathrm e}^{\lambda x} a -x \lambda \left (b +\lambda \right )}{2 \lambda }}\right )}\right ] \\ \end{align*}