Link to actual problem [11095] \[ \boxed {y^{\prime \prime }+y^{\prime } a +b \,{\mathrm e}^{2 a x} y=0} \]
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}^{-x a} \sin \left (\frac {\sqrt {b}\, {\mathrm e}^{x a}}{a}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{x a} y}{\sin \left (\frac {\sqrt {b}\, {\mathrm e}^{x a}}{a}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-x a} \cos \left (\frac {\sqrt {b}\, {\mathrm e}^{x a}}{a}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{x a} y}{\cos \left (\frac {\sqrt {b}\, {\mathrm e}^{x a}}{a}\right )}\right ] \\ \end{align*}