Link to actual problem [9801] \[ \boxed {4 y^{\prime \prime \prime }-8 y^{\prime \prime }-11 y^{\prime }-3 y=-18 \,{\mathrm e}^{x}} \]
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 &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {x}{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= {\mathrm e}^{\frac {x}{2}} y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {x}{2}} x\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x}{2}} y}{x}\right ] \\ \end{align*}
\begin{align*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= -\frac {y}{4}+\frac {{\mathrm e}^{x}}{4}\right ] \\ \\ \end{align*}