Link to actual problem [11511] \[ \boxed {x^{\prime \prime }-2 x^{\prime }+2 x={\mathrm e}^{-t}} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 1] \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 [R &= t, S \left (R \right ) &= \frac {{\mathrm e}^{-t} x}{\sin \left (t \right )}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= t, S \left (R \right ) &= \frac {{\mathrm e}^{-t} x}{\cos \left (t \right )}\right ] \\ \end{align*}
\begin{align*} \\ \\ \end{align*}