Link to actual problem [12743] \[ \boxed {3 y^{\prime \prime }-2 y^{\prime }+4 y=x} \] With initial conditions \begin {align*} [y \left (-1\right ) = 2, y^{\prime }\left (-1\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 &= {\frac {1}{4}}\right ] \\ \left [R &= y-\frac {x}{4}, 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 y\right ] \\ \left [R &= -\frac {\left (-8 y+2 x +1\right ) {\mathrm e}^{-2 x}}{8}, S \left (R \right ) &= x\right ] \\ \end{align*}