Link to actual problem [5152] \[ \boxed {y^{\prime \prime }-2 y^{\prime }+3 y=x^{2}-1} \]
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 &= \frac {4}{9}+\frac {2 x}{3}\right ] \\ \left [R &= y-\frac {x^{2}}{3}-\frac {4 x}{9}, 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}}{2}+\frac {5}{6}+\frac {3 y}{2}\right ] \\ \left [R &= -\frac {\left (9 x^{2}+12 x -27 y-7\right ) {\mathrm e}^{-\frac {3 x}{2}}}{27}, S \left (R \right ) &= x\right ] \\ \end{align*}