Link to actual problem [13677] \[ \boxed {y^{\prime \prime }+2 y^{\prime }-8 y=8 x^{2}-3} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = -3] \end {align*}
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 &= -2 x -\frac {1}{2}\right ] \\ \left [R &= y+x^{2}+\frac {x}{2}, S \left (R \right ) &= x\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 1, \underline {\hspace {1.25 ex}}\eta &= 4 x^{2}-\frac {1}{2}+4 y\right ] \\ \left [R &= \frac {\left (2 x^{2}+x +2 y\right ) {\mathrm e}^{-4 x}}{2}, S \left (R \right ) &= x\right ] \\ \end{align*}