Link to actual problem [9797] \[ \boxed {y^{\prime \prime \prime }+y^{\prime \prime } f \left (x \right )+y^{\prime }+f \left (x \right ) y=0} \]
type detected by program
{"unknown"}
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*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {i {\mathrm e}^{i x} {\mathrm e}^{-2 i x}}{2}\right ] \\ \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-i x} y}{\int {\mathrm e}^{-2 i x} \left (\int {\mathrm e}^{\int \left (i-f \left (x \right )\right )d x}d x \right )d x}\right ] \\ \end{align*}