Link to actual problem [9837] \[ \boxed {x^{3} y^{\prime \prime \prime }-4 y^{\prime \prime } x^{2}+\left (x^{2}+8\right ) x y^{\prime }-2 \left (x^{2}+4\right ) y=0} \]
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*} \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x^{2}}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\cos \left (x \right ) x}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {y}{x \sin \left (x \right )}\right ] \\ \end{align*}