Link to actual problem [9274] \[ \boxed {y^{\prime }-\frac {-32 y x -72 x^{3}+32 x^{2}-32 x +64 y^{3}+48 y^{2} x^{2}-192 y^{2} x +12 x^{4} y-96 y x^{3}+192 y x^{2}+x^{6}-12 x^{5}+48 x^{4}}{64 y+16 x^{2}-64 x +64}=0} \]
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 {x^{6}-12 x^{5}+12 x^{4} y +48 x^{4}-96 x^{3} y +48 x^{2} y^{2}-64 x^{3}+192 x^{2} y -192 x \,y^{2}+64 y^{3}-16 x^{2}+64 x -64 y -64}{16 x^{2}-64 x +64 y +64}\right ] \\ \left [R &= x, S \left (R \right ) &= \int _{}^{y}\frac {16 x^{2}+64 \textit {\_a} -64 x +64}{x^{6}-12 x^{5}+\left (12 \textit {\_a} +48\right ) x^{4}+\left (-96 \textit {\_a} -64\right ) x^{3}+\left (48 \textit {\_a}^{2}+192 \textit {\_a} -16\right ) x^{2}+\left (-192 \textit {\_a}^{2}+64\right ) x +64 \textit {\_a}^{3}-64 \textit {\_a} -64}d \textit {\_a}\right ] \\ \end{align*}