Link to actual problem [6500] \[ \boxed {y^{\prime \prime }+y^{\prime }-6 y=t} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, 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 [\underline {\hspace {1.25 ex}}\xi &= 1, \underline {\hspace {1.25 ex}}\eta &= -{\frac {1}{6}}\right ] \\ \left [R &= y+\frac {t}{6}, S \left (R \right ) &= t\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= \frac {\left (36 y+6 t +1\right ) {\mathrm e}^{-6 t}}{36}, S \left (R \right ) &= t\right ] \\ \end{align*}