Link to actual problem [11091] \[ \boxed {y^{\prime \prime }-\left (a^{2} {\mathrm e}^{2 x}+a \left (1+2 b \right ) {\mathrm e}^{x}+b^{2}\right ) y=0} \]
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 (a^{-2 b -1} b -\frac {a^{-2 b -1}}{2}\right ) {\mathrm e}^{-x} \operatorname {WhittakerM}\left (-b +1, \frac {1}{2}-b , 2 a \,{\mathrm e}^{x}\right )-a^{-2 b} \operatorname {WhittakerM}\left (-b , \frac {1}{2}-b , 2 a \,{\mathrm e}^{x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {2 a^{2 b} a \,{\mathrm e}^{x} y}{\left (2 b -1\right ) \operatorname {WhittakerM}\left (-b +1, \frac {1}{2}-b , 2 a \,{\mathrm e}^{x}\right )-2 \operatorname {WhittakerM}\left (-b , \frac {1}{2}-b , 2 a \,{\mathrm e}^{x}\right ) a \,{\mathrm e}^{x}}\right ] \\ \end{align*}