Link to actual problem [14854] \[ \boxed {y^{\prime \prime \prime }+3 y^{\prime \prime }-9 y^{\prime }+5 y={\mathrm e}^{t}} \]
type detected by program
{"higher_order_linear_constant_coefficients_ODE"}
type detected by Maple
[[_3rd_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 [R &= t, S \left (R \right ) &= \frac {{\mathrm e}^{-t} y}{t}\right ] \\ \end{align*}
\begin{align*} \\ \\ \end{align*}
\begin{align*} \\ \left [R &= t, S \left (R \right ) &= -\frac {\ln \left ({\mathrm e}^{t} t^{2}-12 y\right )}{12}\right ] \\ \end{align*}