Link to actual problem [12028] \[ \boxed {x^{\prime \prime }-4 x=t^{2}} \]
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 {t}{2}\right ] \\ \left [R &= x+\frac {t^{2}}{4}, S \left (R \right ) &= t\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= t^{2}+\frac {1}{2}+4 x\right ] \\ \left [R &= t, S \left (R \right ) &= \frac {\ln \left (2 t^{2}+8 x+1\right )}{4}\right ] \\ \end{align*}