Link to actual problem [11098] \[ \boxed {y^{\prime \prime }-y^{\prime }+\left ({\mathrm e}^{3 \lambda x} a +b \,{\mathrm e}^{2 \lambda x}+\frac {1}{4}-\frac {\lambda ^{2}}{4}\right ) y=0} \]
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 &= {\mathrm e}^{-\frac {x \left (\lambda -1\right )}{2}} \operatorname {AiryAi}\left (-\frac {{\mathrm e}^{\lambda x} a +b}{\lambda ^{\frac {2}{3}} a^{\frac {2}{3}}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x \left (\lambda -1\right )}{2}} y}{\operatorname {AiryAi}\left (-\frac {{\mathrm e}^{\lambda x} a +b}{\lambda ^{\frac {2}{3}} a^{\frac {2}{3}}}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {x \left (\lambda -1\right )}{2}} \operatorname {AiryBi}\left (-\frac {{\mathrm e}^{\lambda x} a +b}{\lambda ^{\frac {2}{3}} a^{\frac {2}{3}}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x \left (\lambda -1\right )}{2}} y}{\operatorname {AiryBi}\left (-\frac {{\mathrm e}^{\lambda x} a +b}{\lambda ^{\frac {2}{3}} a^{\frac {2}{3}}}\right )}\right ] \\ \end{align*}