Link to actual problem [2830] \[ \boxed {y^{\prime \prime \prime }+9 y^{\prime \prime }+24 y^{\prime }+16 y=8 \,{\mathrm e}^{-x}+1} \]
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 [\underline {\hspace {1.25 ex}}\xi &= 1, \underline {\hspace {1.25 ex}}\eta &= -y +\frac {1}{16}\right ] \\ \left [R &= \frac {\left (16 y-1\right ) {\mathrm e}^{x}}{16}, S \left (R \right ) &= x\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= -\frac {9 y}{8}+\frac {9}{128}+x \,{\mathrm e}^{-x}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {8 \ln \left (-144 \,{\mathrm e}^{x} y+9 \,{\mathrm e}^{x}+128 x \right )}{9}\right ] \\ \end{align*}