Link to actual problem [9272] \[ \boxed {y^{\prime }-\frac {-32 y x +16 x^{3}+16 x^{2}-32 x -64 y^{3}+48 y^{2} x^{2}+96 y^{2} x -12 x^{4} y-48 y x^{3}-48 y x^{2}+x^{6}+6 x^{5}+12 x^{4}}{-64 y+16 x^{2}+32 x -64}=0} \]
type detected by program
{"first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, `2nd type`, `class C`]]
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 {\left (x^{2}+2 x -4 y +4\right ) \left (x^{4}+4 x^{3}-8 x^{2} y -16 x y +16 y^{2}-8 x +16 y +8\right )}{16 x^{2}+32 x -64 y -64}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {2 \ln \left (x^{4}-8 x^{2} y+4 x^{3}+16 y^{2}-16 x y+16 y-8 x +8\right )}{5}+\frac {2 \arctan \left (-2 y+\frac {x^{2}}{2}+x -1\right )}{5}+\frac {4 \ln \left (-x^{2}-2 x +4 y-4\right )}{5}\right ] \\ \end{align*}