Link to actual problem [1782] \[ \boxed {y^{\prime \prime }+y^{\prime }+{\mathrm e}^{-t} y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = 5] \end {align*}
With the expansion point for the power series method at \(t = 0\).
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 &= {\mathrm e}^{-\frac {t}{2}} \operatorname {BesselJ}\left (1, 2 \,{\mathrm e}^{-\frac {t}{2}}\right )\right ] \\ \left [R &= t, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {t}{2}} y}{\operatorname {BesselJ}\left (1, 2 \,{\mathrm e}^{-\frac {t}{2}}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {t}{2}} \operatorname {BesselY}\left (1, 2 \,{\mathrm e}^{-\frac {t}{2}}\right )\right ] \\ \left [R &= t, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {t}{2}} y}{\operatorname {BesselY}\left (1, 2 \,{\mathrm e}^{-\frac {t}{2}}\right )}\right ] \\ \end{align*}