Link to actual problem [15309] \[ \boxed {y^{\prime \prime \prime }-y^{\prime \prime }+y^{\prime }-y=x^{2}+x} \]
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 [R &= x, S \left (R \right ) &= \frac {y}{\sin \left (x \right )}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\cos \left (x \right )}\right ] \\ \end{align*}
\begin{align*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 1, \underline {\hspace {1.25 ex}}\eta &= \frac {2 x^{2}}{3}-\frac {7}{3}+\frac {2 y}{3}\right ] \\ \left [R &= \left (x^{2}+3 x +y+1\right ) {\mathrm e}^{-\frac {2 x}{3}}, S \left (R \right ) &= x\right ] \\ \end{align*}