Link to actual problem [15290] \[ \boxed {y^{\prime \prime }+4 y^{\prime }+4 y=8 \,{\mathrm e}^{-2 x}} \]
type detected by program
{"kovacic", "second_order_linear_constant_coeff", "linear_second_order_ode_solved_by_an_integrating_factor"}
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 &= -\frac {x}{2}-\frac {1}{2}, \underline {\hspace {1.25 ex}}\eta &= x y\right ] \\ \left [R &= \frac {y \,{\mathrm e}^{2 x}}{x^{2}+2 x +1}, S \left (R \right ) &= -2 \ln \left (-x -1\right )\right ] \\ \end{align*}