Link to actual problem [3217] \[ \boxed {\left (x +1\right ) \left (y^{\prime }+y^{2}\right )-y=0} \]
type detected by program
{}
type detected by Maple
[[_1st_order, _with_linear_symmetries], _rational, _Bernoulli]
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 {y^{2}}{1+x}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {1+x}{y}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {y \left (x^{2} y +2 x y -2 x -2\right )}{1+x}\right ] \\ \left [R &= x, S \left (R \right ) &= \left (1+x \right ) \left (\frac {x \left (2+x \right ) \ln \left (x^{2} y+2 x y-2 x -2\right )}{\left (2 x +2\right ) \left (x^{2}+2 x \right )}-\frac {\ln \left (y\right )}{2 x +2}\right )\right ] \\ \end{align*}