Link to actual problem [12793] \[ \boxed {y^{\prime \prime }-9 y=x +2} \] With initial conditions \begin {align*} [y \left (0\right ) = -1, y^{\prime }\left (0\right ) = 1] \end {align*}
type detected by program
{"second_order_laplace", "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}{9}}\right ] \\ \left [R &= y+\frac {x}{9}, 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}{18}+\frac {y}{2}\right ] \\ \left [R &= \frac {\left (9 y+x +2\right ) {\mathrm e}^{-\frac {x}{2}}}{9}, S \left (R \right ) &= x\right ] \\ \end{align*}