Link to actual problem [13189] \[ \boxed {y^{\prime \prime }+4 y=-3 t^{2}+2 t +3} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = 0] \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 [\underline {\hspace {1.25 ex}}\xi &= 1, \underline {\hspace {1.25 ex}}\eta &= -\frac {3 t}{2}+\frac {1}{2}\right ] \\ \left [R &= y+\frac {3 t^{2}}{4}-\frac {t}{2}, S \left (R \right ) &= t\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 1, \underline {\hspace {1.25 ex}}\eta &= \frac {1}{3} t^{2}-\frac {31}{18} t +\frac {4}{9} y\right ] \\ \left [R &= \frac {\left (6 t^{2}-4 t +8 y-9\right ) {\mathrm e}^{-\frac {4 t}{9}}}{8}, S \left (R \right ) &= t\right ] \\ \end{align*}