Link to actual problem [14417] \[ \boxed {-y+\left (-t +y\right ) y^{\prime }=-t^{2}} \]
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}{-y +t}\right ] \\ \left [R &= t, S \left (R \right ) &= -\frac {y^{2}}{2}+t y\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {2 t^{3}-6 t y +3 y^{2}}{-2 y +2 t}\right ] \\ \left [R &= t, S \left (R \right ) &= -\frac {\ln \left (2 t^{3}-6 t y+3 y^{2}\right )}{3}\right ] \\ \end{align*}