Link to actual problem [14555] \[ \boxed {y^{\prime \prime }+y^{\prime }-6 y=3 t} \] With initial conditions \begin {align*} \left [y \left (0\right ) = {\frac {23}{12}}, y^{\prime }\left (0\right ) = -{\frac {3}{2}}\right ] \end {align*}
type detected by program
{"kovacic", "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 {\ln \left (6 t +12 y+1\right )}{12}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= \frac {\left (6 t +12 y+1\right ) {\mathrm e}^{-6 t}}{12}, S \left (R \right ) &= t\right ] \\ \end{align*}