Link to actual problem [5656] \[ \boxed {y^{\prime \prime }+\left ({\mathrm e}^{-2 x}-\frac {1}{9}\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second_order_bessel_ode_form_A", "second order series method. Ordinary point", "second order series method. Taylor series method"}
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 (\frac {1}{3}, {\mathrm e}^{-x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {BesselJ}\left (\frac {1}{3}, {\mathrm e}^{-x}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \operatorname {BesselY}\left (\frac {1}{3}, {\mathrm e}^{-x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\operatorname {BesselY}\left (\frac {1}{3}, {\mathrm e}^{-x}\right )}\right ] \\ \end{align*}