Link to actual problem [686] \[ \boxed {4 y^{\prime \prime }-4 y^{\prime }+y=16 \,{\mathrm e}^{\frac {t}{2}}} \]
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*} \left [\underline {\hspace {1.25 ex}}\xi &= 1, \underline {\hspace {1.25 ex}}\eta &= \frac {y}{2}\right ] \\ \left [R &= y \,{\mathrm e}^{-\frac {t}{2}}, S \left (R \right ) &= t\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= \frac {y \,{\mathrm e}^{-\frac {t}{2}}}{t^{2}-8 t +16}, S \left (R \right ) &= \frac {\ln \left (t -4\right )}{2}\right ] \\ \end{align*}