Link to actual problem [12796] \[ \boxed {y^{\prime \prime }-2 y^{\prime }+2 y=-x^{2}+1} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \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 [R &= y+\frac {x^{2}}{2}+x, 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 -1\right ] \\ \left [R &= \frac {\left (x^{2}+2 x +2 y\right ) {\mathrm e}^{-x}}{2}, S \left (R \right ) &= x\right ] \\ \end{align*}