Link to actual problem [12803] \[ \boxed {y^{\prime \prime }-y^{\prime }-2 y=x^{2}} \] With initial conditions \begin {align*} \left [y \left (0\right ) = {\frac {11}{4}}, y^{\prime }\left (0\right ) = {\frac {1}{2}}\right ] \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}{2}-x\right ] \\ \left [R &= y+\frac {x^{2}}{2}-\frac {x}{2}, S \left (R \right ) &= x\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= \frac {\left (2 x^{2}-2 x +4 y+3\right ) {\mathrm e}^{2 x}}{4}, S \left (R \right ) &= x\right ] \\ \end{align*}