Link to actual problem [1153] \[ \boxed {x^{2} \left (y^{\prime }+y^{2}\right )+y x=\frac {1}{4}-x^{2}} \]
type detected by program
{"riccati"}
type detected by Maple
[_rational, [_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 &= 0, \underline {\hspace {1.25 ex}}\eta &= 1+\frac {\left (2 x y +1\right )^{2}}{4 x^{2}}\right ] \\ \left [R &= x, S \left (R \right ) &= \arctan \left (\frac {8 x^{2} y+4 x}{8 x^{2}}\right )\right ] \\ \end{align*}