Link to actual problem [2005] \[ \boxed {y^{2}+\left (2 y x -y^{2}\right ) y^{\prime }=-1} \] With initial conditions \begin {align*} [y \left (0\right ) = -1] \end {align*}
type detected by program
{"exact"}
type detected by Maple
[_exact, _rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]
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}{y \left (2 x -y \right )}\right ] \\ \left [R &= x, S \left (R \right ) &= x y^{2}-\frac {y^{3}}{3}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= -\frac {3 x \,y^{2}-y^{3}+3 x}{y \left (2 x -y \right )}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {\ln \left (-3 x y^{2}+y^{3}-3 x \right )}{3}\right ] \\ \end{align*}