Link to actual problem [11257] \[ \boxed {y^{\prime \prime \prime }-3 y^{\prime \prime }-y^{\prime }+3 y=x^{2}} \]
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*} \\ \\ \end{align*}
\begin{align*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 1, \underline {\hspace {1.25 ex}}\eta &= \frac {2}{9}+\frac {2 x}{3}\right ] \\ \left [R &= y-\frac {x^{2}}{3}-\frac {2 x}{9}, S \left (R \right ) &= x\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= -\frac {\left (9 x^{2}+6 x -27 y+20\right ) {\mathrm e}^{-3 x}}{27}, S \left (R \right ) &= x\right ] \\ \end{align*}