Link to actual problem [11259] \[ \boxed {y^{\prime \prime \prime }-4 y^{\prime \prime }+5 y^{\prime }-2 y=x} \]
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*} \left [\underline {\hspace {1.25 ex}}\xi &= 1, \underline {\hspace {1.25 ex}}\eta &= -{\frac {1}{2}}\right ] \\ \left [R &= y+\frac {x}{2}, 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 {x}{2}+y +\frac {5}{4}\right ] \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 1, \underline {\hspace {1.25 ex}}\eta &= \frac {2 y}{5}+\frac {x}{5}\right ] \\ \left [R &= \frac {\left (2 x +4 y+5\right ) {\mathrm e}^{-\frac {2 x}{5}}}{4}, S \left (R \right ) &= x\right ] \\ \end{align*}