Link to actual problem [6863] \[ \boxed {y^{\prime \prime }+3 y^{\prime }+2 y=12 x^{2}} \]
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*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 1, \underline {\hspace {1.25 ex}}\eta &= 4 x^{2}-4-\frac {2 y}{3}\right ] \\ \left [R &= -\left (6 x^{2}-18 x -y+21\right ) {\mathrm e}^{\frac {2 x}{3}}, S \left (R \right ) &= x\right ] \\ \end{align*}