Link to actual problem [14522] \[ \boxed {y^{\prime \prime }-2 y^{\prime }+y=t^{2}} \]
type detected by program
{"kovacic", "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*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 1, \underline {\hspace {1.25 ex}}\eta &= -\frac {t^{2}}{2}+\frac {y}{2}+1\right ] \\ \left [R &= -\left (t^{2}+4 t -y+6\right ) {\mathrm e}^{-\frac {t}{2}}, S \left (R \right ) &= t\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= -\frac {\left (t^{2}+4 t -y+6\right ) {\mathrm e}^{-t}}{t -3}, S \left (R \right ) &= \ln \left (t -3\right )\right ] \\ \end{align*}