Link to actual problem [5852] \[ \boxed {y^{\prime \prime }-2 y^{\prime }-3 y=3 x +1} \]
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 [R &= x, S \left (R \right ) &= -\frac {\ln \left (-3 x -3 y+1\right )}{3}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= \frac {\left (3 x +3 y-1\right ) {\mathrm e}^{3 x}}{3}, S \left (R \right ) &= x\right ] \\ \end{align*}