Link to actual problem [13739] \[ \boxed {y^{\prime \prime }-5 y^{\prime }+6 y=x^{2}} \]
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*} \left [\underline {\hspace {1.25 ex}}\xi &= 1, \underline {\hspace {1.25 ex}}\eta &= \frac {5}{18}+\frac {x}{3}\right ] \\ \left [R &= y-\frac {x^{2}}{6}-\frac {5 x}{18}, 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 {x^{2}}{5}+\frac {1}{15}+\frac {6 y}{5}\right ] \\ \left [R &= -\frac {\left (18 x^{2}+30 x -108 y+19\right ) {\mathrm e}^{-\frac {6 x}{5}}}{108}, S \left (R \right ) &= x\right ] \\ \end{align*}