Link to actual problem [11113] \[ \boxed {y^{\prime \prime }+\left (a \,{\mathrm e}^{x}+b \right ) y^{\prime }+\left (c \left (a -c \right ) {\mathrm e}^{2 x}+\left (a k +b c -2 c k +c \right ) {\mathrm e}^{x}+k \left (b -k \right )\right ) y=0} \]
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 (-1+b -2 k \right ) \left (-2 c +a \right )^{-b +2 k -1} \operatorname {WhittakerM}\left (-\frac {b}{2}+k +1, -\frac {b}{2}+k +\frac {1}{2}, \left (-2 c +a \right ) {\mathrm e}^{x}\right ) {\mathrm e}^{-\frac {a \,{\mathrm e}^{x}}{2}-\frac {x \left (b +2\right )}{2}}-\left (-2 c +a \right )^{-b +2 k} \operatorname {WhittakerM}\left (-\frac {b}{2}+k , -\frac {b}{2}+k +\frac {1}{2}, \left (-2 c +a \right ) {\mathrm e}^{x}\right ) {\mathrm e}^{-\frac {a \,{\mathrm e}^{x}}{2}-\frac {b x}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {\left (-2 c +a \right )^{-2 k} \left (-2 c +a \right )^{b} \left (-2 c +a \right ) {\mathrm e}^{\frac {a \,{\mathrm e}^{x}}{2}} {\mathrm e}^{\frac {x \left (b +2\right )}{2}} {\mathrm e}^{\frac {b x}{2}} y}{-{\mathrm e}^{\frac {b x}{2}} \left (-1+b -2 k \right ) \operatorname {WhittakerM}\left (-\frac {b}{2}+k +1, -\frac {b}{2}+k +\frac {1}{2}, \left (-2 c +a \right ) {\mathrm e}^{x}\right )+{\mathrm e}^{\frac {x \left (b +2\right )}{2}} \operatorname {WhittakerM}\left (-\frac {b}{2}+k , -\frac {b}{2}+k +\frac {1}{2}, \left (-2 c +a \right ) {\mathrm e}^{x}\right ) \left (-2 c +a \right )}\right ] \\ \end{align*}