Link to actual problem [6864] \[ \boxed {y^{\prime \prime }+3 y^{\prime }+2 y=x^{2}+2 x +1} \]
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 [\underline {\hspace {1.25 ex}}\xi &= 1, \underline {\hspace {1.25 ex}}\eta &= x -\frac {1}{2}\right ] \\ \left [R &= y-\frac {x^{2}}{2}+\frac {x}{2}, S \left (R \right ) &= x\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= -\frac {\left (2 x^{2}-2 x -4 y+3\right ) {\mathrm e}^{2 x}}{4}, S \left (R \right ) &= x\right ] \\ \end{align*}