Link to actual problem [11466] \[ \boxed {x^{\prime \prime }-3 x^{\prime }-4 x=2 t^{2}} \]
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}{4}-t\right ] \\ \left [R &= x+\frac {t^{2}}{2}-\frac {3 t}{4}, 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 {2 t^{2}}{3}-\frac {1}{3}-\frac {4 x}{3}\right ] \\ \left [R &= \frac {\left (8 t^{2}-12 t +16 x+13\right ) {\mathrm e}^{\frac {4 t}{3}}}{16}, S \left (R \right ) &= t\right ] \\ \end{align*}