Link to actual problem [13193] \[ \boxed {y^{\prime \prime }+4 y=t -\frac {1}{20} t^{2}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, 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 {1}{4}-\frac {t}{40}\right ] \\ \left [R &= \frac {t^{2}}{80}-\frac {t}{4}+y, S \left (R \right ) &= t\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {1}{80} t^{2}-\frac {1}{160}-\frac {1}{4} t +y\right ] \\ \\ \end{align*}