Link to actual problem [6508] \[ \boxed {y^{\prime \prime }-y^{\prime }+y=3 \,{\mathrm e}^{-t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = 2] \end {align*}
type detected by program
{"second_order_laplace", "second_order_linear_constant_coeff"}
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*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right )\right ] \\ \left [R &= t, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {t}{2}} y}{\sin \left (\frac {\sqrt {3}\, 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}} \cos \left (\frac {\sqrt {3}\, t}{2}\right )\right ] \\ \left [R &= t, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {t}{2}} y}{\cos \left (\frac {\sqrt {3}\, t}{2}\right )}\right ] \\ \end{align*}
\begin{align*} \\ \\ \end{align*}