Link to actual problem [5151] \[ \boxed {y^{\prime \prime }-4 y^{\prime }+3 y=x +{\mathrm e}^{2 x}} \]
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*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 1, \underline {\hspace {1.25 ex}}\eta &= -\frac {2 x}{3}+2 y -\frac {5}{9}\right ] \\ \left [R &= -\frac {\left (-9 y+3 x +4\right ) {\mathrm e}^{-2 x}}{9}, S \left (R \right ) &= x\right ] \\ \end{align*}