Link to actual problem [238] \[ \boxed {y^{\prime \prime }-2 y^{\prime }+2 y=x +1} \] With initial conditions \begin {align*} [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = 0] \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}{2}}\right ] \\ \left [R &= y-\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 &= \frac {y}{2}-\frac {x}{4}\right ] \\ \left [R &= -\frac {\left (-2 y+x +2\right ) {\mathrm e}^{-\frac {x}{2}}}{2}, S \left (R \right ) &= x\right ] \\ \end{align*}