Link to actual problem [11774] \[ \boxed {y^{\prime \prime }-3 y^{\prime }+8 y=4 x^{2}} \]
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 {3}{8}+x\right ] \\ \left [R &= y-\frac {x^{2}}{2}-\frac {3 x}{8}, 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 {4 x^{2}}{3}+\frac {1}{3}+\frac {8 y}{3}\right ] \\ \left [R &= -\frac {\left (32 x^{2}+24 x -64 y+1\right ) {\mathrm e}^{-\frac {8 x}{3}}}{64}, S \left (R \right ) &= x\right ] \\ \end{align*}