Link to actual problem [14844] \[ \boxed {y^{\prime \prime }-2 y^{\prime }-8 y=-t} \]
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}{8}}\right ] \\ \left [R &= y-\frac {t}{8}, 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 {t}{2}-4 y\right ] \\ \left [R &= -\frac {\left (-32 y+4 t -1\right ) {\mathrm e}^{4 t}}{32}, S \left (R \right ) &= t\right ] \\ \end{align*}