Link to actual problem [11097] \[ \boxed {y^{\prime \prime }+y^{\prime } a +\left (b \,{\mathrm e}^{\lambda x}+c \right ) y=0} \]
type detected by program
{"second_order_bessel_ode_form_A"}
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 a}{2}} \operatorname {BesselJ}\left (\frac {\sqrt {a^{2}-4 c}}{\lambda }, \frac {2 \sqrt {b}\, {\mathrm e}^{\frac {\lambda x}{2}}}{\lambda }\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x a}{2}} y}{\operatorname {BesselJ}\left (\frac {\sqrt {a^{2}-4 c}}{\lambda }, \frac {2 \sqrt {b}\, {\mathrm e}^{\frac {\lambda x}{2}}}{\lambda }\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {x a}{2}} \operatorname {BesselY}\left (\frac {\sqrt {a^{2}-4 c}}{\lambda }, \frac {2 \sqrt {b}\, {\mathrm e}^{\frac {\lambda x}{2}}}{\lambda }\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x a}{2}} y}{\operatorname {BesselY}\left (\frac {\sqrt {a^{2}-4 c}}{\lambda }, \frac {2 \sqrt {b}\, {\mathrm e}^{\frac {\lambda x}{2}}}{\lambda }\right )}\right ] \\ \end{align*}