Link to actual problem [6672] \[ \boxed {y^{\prime \prime }-6 y^{\prime }+9 y=t} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
type detected by program
{"second_order_laplace", "second_order_linear_constant_coeff", "linear_second_order_ode_solved_by_an_integrating_factor"}
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}{9}}\right ] \\ \left [R &= y-\frac {t}{9}, 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 {3 y}{2}-\frac {t}{6}\right ] \\ \left [R &= -\frac {\left (-27 y+3 t +2\right ) {\mathrm e}^{-\frac {3 t}{2}}}{27}, S \left (R \right ) &= t\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= -\frac {\left (-27 y+3 t +2\right ) {\mathrm e}^{-3 t}}{27 \left (3 t -2\right )}, S \left (R \right ) &= -\frac {\ln \left (-3 t +2\right )}{3}\right ] \\ \end{align*}