Link to actual problem [11089] \[ \boxed {y^{\prime \prime }+\left (a \,{\mathrm e}^{x}-b \right ) y=0} \]
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 &= \operatorname {BesselJ}\left (2 \sqrt {b}, 2 \sqrt {a}\, {\mathrm e}^{\frac {x}{2}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {BesselJ}\left (2 \sqrt {b}, 2 \sqrt {a}\, {\mathrm e}^{\frac {x}{2}}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {BesselY}\left (2 \sqrt {b}, 2 \sqrt {a}\, {\mathrm e}^{\frac {x}{2}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {BesselY}\left (2 \sqrt {b}, 2 \sqrt {a}\, {\mathrm e}^{\frac {x}{2}}\right )}\right ] \\ \end{align*}