Link to actual problem [9280] \[ \boxed {y^{\prime }-\frac {2 x^{2} \cos \left (x \right )+2 x^{3} \sin \left (x \right )-2 x \sin \left (x \right )+2 x +2 y^{2} x^{2}-4 y \sin \left (x \right ) x +4 y \cos \left (x \right ) x^{2}+4 x y+3-\cos \left (2 x \right )-2 \sin \left (2 x \right ) x -4 \sin \left (x \right )+\cos \left (2 x \right ) x^{2}+x^{2}+4 x \cos \left (x \right )}{2 x^{3}}=0} \]
type detected by program
{"riccati"}
type detected by Maple
[[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\).\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= x, \underline {\hspace {1.25 ex}}\eta &= \frac {\sin \left (x \right ) x^{2}+\cos \left (x \right ) x -\sin \left (x \right )+1}{x}\right ] \\ \left [R &= y+\frac {-\sin \left (x \right )+1}{x}+\cos \left (x \right ), S \left (R \right ) &= \ln \left (x \right )\right ] \\ \end{align*}