Link to actual problem [12518] \[ \boxed {s^{\prime \prime }-a^{2} s=1+t} \]
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}{a^{2}}\right ] \\ \left [R &= s+\frac {t}{a^{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 &= s +\frac {t}{a^{2}}\right ] \\ \left [R &= \frac {\left (a^{2} s+t +1\right ) {\mathrm e}^{-t}}{a^{2}}, S \left (R \right ) &= t\right ] \\ \end{align*}