Link to actual problem [9267] \[ \boxed {y^{\prime }-y^{2}-\frac {y x^{2}}{4}+y x -y^{3}+\frac {3 y^{2} x^{2}}{4}+\frac {3 y^{2} x}{2}-\frac {3 x^{4} y}{16}-\frac {3 y x^{3}}{4}=1+\frac {1}{2} x -\frac {1}{8} x^{4}+\frac {1}{8} x^{3}+\frac {1}{4} x^{2}-\frac {1}{64} x^{6}-\frac {3}{32} x^{5}} \]
type detected by program
{"abelFirstKind", "first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], _Abel]
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}{2}+\frac {\left (x^{2}+2 x -4 y -4\right ) \left (x^{2}+2 x -4 y \right )^{2}}{64}\right ] \\ \left [R &= x, S \left (R \right ) &= \int _{}^{y}\frac {1}{-\frac {1}{2}+\frac {x^{6}}{64}+\frac {3 x^{5}}{32}+\frac {\left (2-3 \textit {\_a} \right ) x^{4}}{16}+\frac {\left (-1-6 \textit {\_a} \right ) x^{3}}{8}+\frac {\left (3 \textit {\_a}^{2}-\textit {\_a} -1\right ) x^{2}}{4}+\frac {\left (3 \textit {\_a}^{2}+2 \textit {\_a} \right ) x}{2}-\textit {\_a}^{3}-\textit {\_a}^{2}}d \textit {\_a}\right ] \\ \end{align*}