Link to actual problem [12315] \[ \boxed {y^{\prime \prime }-2 y^{\prime }+5 y=2+t} \] With initial conditions \begin {align*} [y \left (0\right ) = 4, 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}{5}}\right ] \\ \left [R &= y-\frac {t}{5}, S \left (R \right ) &= t\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 1, \underline {\hspace {1.25 ex}}\eta &= -\frac {t}{12}+\frac {5 y}{12}\right ] \\ \left [R &= -\frac {\left (-25 y+5 t +12\right ) {\mathrm e}^{-\frac {5 t}{12}}}{25}, S \left (R \right ) &= t\right ] \\ \end{align*}