Link to actual problem [6506] \[ \boxed {y^{\prime \prime }+3 y^{\prime }-2 y=-6 \,{\mathrm e}^{\pi -t}} \] With initial conditions \begin {align*} [y \left (\pi \right ) = 1, y^{\prime }\left (\pi \right ) = 4] \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 &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{\frac {\left (-3+\sqrt {17}\right ) t}{2}}\right ] \\ \left [R &= t, S \left (R \right ) &= {\mathrm e}^{-\frac {\left (-3+\sqrt {17}\right ) t}{2}} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {\left (3+\sqrt {17}\right ) t}{2}}\right ] \\ \left [R &= t, S \left (R \right ) &= {\mathrm e}^{\frac {\left (3+\sqrt {17}\right ) t}{2}} y\right ] \\ \end{align*}
\begin{align*} \\ \\ \end{align*}