Link to actual problem [6380] \[ \boxed {y^{\prime \prime }-y^{\prime }+4 y=x} \] With initial conditions \begin {align*} [y \left (1\right ) = 2, y^{\prime }\left (1\right ) = 1] \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 [R &= -\frac {\left (-16 y+4 x +1\right ) {\mathrm e}^{-4 x}}{16}, S \left (R \right ) &= x\right ] \\ \end{align*}