Link to actual problem [2750] \[ \boxed {y^{\prime \prime \prime }+2 y^{\prime \prime }-5 y^{\prime }-6 y=4 x^{2}} \]
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 {4 x}{3}+\frac {10}{9}\right ] \\ \left [R &= y+\frac {2 x^{2}}{3}-\frac {10 x}{9}, S \left (R \right ) &= x\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 1, \underline {\hspace {1.25 ex}}\eta &= -\frac {4 x^{2}}{5}-\frac {8}{15}-\frac {6 y}{5}\right ] \\ \left [R &= \frac {\left (18 x^{2}-30 x +27 y+37\right ) {\mathrm e}^{\frac {6 x}{5}}}{27}, S \left (R \right ) &= x\right ] \\ \end{align*}