Link to actual problem [3722] \[ \boxed {2 \left (x +y\right ) y^{\prime }+2 y=-x^{2}} \]
type detected by program
{"exact", "differentialType"}
type detected by Maple
[_exact, _rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], [_Abel, `2nd type`, `class A`]]
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 &= \frac {1}{2 x +2 y}\right ] \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {x^{3}+6 x y +3 y^{2}}{2 x +2 y}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\ln \left (x^{3}+6 x y+3 y^{2}\right )}{3}\right ] \\ \end{align*}