Link to actual problem [12517] \[ \boxed {y^{\prime \prime }-7 y^{\prime }+12 y=x} \]
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 [\underline {\hspace {1.25 ex}}\xi &= 1, \underline {\hspace {1.25 ex}}\eta &= {\frac {1}{12}}\right ] \\ \left [R &= y-\frac {x}{12}, S \left (R \right ) &= x\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 1, \underline {\hspace {1.25 ex}}\eta &= \frac {12 y}{7}-\frac {x}{7}\right ] \\ \left [R &= -\frac {\left (-144 y+12 x +7\right ) {\mathrm e}^{-\frac {12 x}{7}}}{144}, S \left (R \right ) &= x\right ] \\ \end{align*}