Link to actual problem [4827] \[ \boxed {5 y^{\prime \prime }+6 y^{\prime }+2 y=x^{2}+6 x} \]
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*} \\ \left [R &= -\frac {x^{2}}{2}+y, S \left (R \right ) &= x\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= -\frac {x^{2}}{2}+\frac {5}{2}+y\right ] \\ \\ \end{align*}