Link to actual problem [9284] \[ \boxed {y^{\prime }-y^{2}-\frac {x^{2} y}{2}-a x y-y^{3}-\frac {3 y^{2} x^{2}}{4}-\frac {3 y^{2} a x}{2}-\frac {3 y x^{4}}{16}-\frac {3 y a \,x^{3}}{4}-\frac {3 y a^{2} x^{2}}{4}=-\frac {1}{2} x +1+\frac {1}{16} x^{4}+\frac {1}{4} a \,x^{3}+\frac {1}{4} a^{2} x^{2}+\frac {1}{64} x^{6}+\frac {3}{32} a \,x^{5}+\frac {3}{16} a^{2} x^{4}+\frac {1}{8} a^{3} x^{3}} \]
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}{8} a^{3} x^{3}+\frac {3}{16} a^{2} x^{4}+\frac {3}{32} x^{5} a +\frac {1}{64} x^{6}+\frac {3}{4} a^{2} x^{2} y +\frac {3}{4} a \,x^{3} y +\frac {3}{16} x^{4} y +\frac {1}{4} a^{2} x^{2}+\frac {1}{4} a \,x^{3}+\frac {3}{2} a x \,y^{2}+\frac {1}{16} x^{4}+\frac {3}{4} x^{2} y^{2}+a x y +\frac {1}{2} x^{2} y +y^{3}+y^{2}+\frac {1}{2} a +1\right ] \\ \left [R &= x, S \left (R \right ) &= \int _{}^{y}\frac {1}{1+\frac {x^{6}}{64}+\frac {3 x^{5} a}{32}+\frac {\left (3 a^{2}+3 \textit {\_a} +1\right ) x^{4}}{16}+\frac {a \left (a^{2}+6 \textit {\_a} +2\right ) x^{3}}{8}+\frac {\left (\left (3 \textit {\_a} +1\right ) a^{2}+3 \textit {\_a}^{2}+2 \textit {\_a} \right ) x^{2}}{4}+\frac {3 a \textit {\_a} \left (\textit {\_a} +\frac {2}{3}\right ) x}{2}+\textit {\_a}^{3}+\textit {\_a}^{2}+\frac {a}{2}}d \textit {\_a}\right ] \\ \end{align*}
My program’s symgen result This shows my program’s found \(\xi ,\eta \) and the corresponding ODE in canonical coordinates \(R,S\).\begin{align*} \xi &= -2 \\ \eta &=x +a \\ \frac {dS}{dR} &= -\frac {1}{2 R^{3}+2 R^{2}+a +2} \\ \end{align*}