Link to actual problem [9283] \[ \boxed {y^{\prime }-y^{2}-\frac {y a \,x^{2}}{2}-b x y-y^{3}-\frac {3 a \,x^{2} y^{2}}{4}-\frac {3 y^{2} b x}{2}-\frac {3 y a^{2} x^{4}}{16}-\frac {3 y a \,x^{3} b}{4}-\frac {3 b^{2} x^{2} y}{4}=-\frac {1}{2} x a +1+\frac {1}{16} a^{2} x^{4}+\frac {1}{4} x^{3} a b +\frac {1}{4} x^{2} b^{2}+\frac {1}{64} a^{3} x^{6}+\frac {3}{32} a^{2} x^{5} b +\frac {3}{16} a \,x^{4} b^{2}+\frac {1}{8} b^{3} x^{3}} \]
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}{64} b x y +\frac {3}{256} b^{2} x^{2} y +\frac {1}{128} a \,x^{2} y +\frac {3}{128} b x \,y^{2}+\frac {3}{256} a b \,x^{3} y +\frac {3}{256} a \,x^{2} y^{2}+\frac {3}{1024} a^{2} x^{4} y +\frac {1}{4096} a^{3} x^{6}+\frac {1}{128} b +\frac {1}{64}+\frac {1}{256} b^{2} x^{2}+\frac {1}{64} y^{2}+\frac {1}{512} b^{3} x^{3}+\frac {1}{256} a b \,x^{3}+\frac {1}{64} y^{3}+\frac {3}{1024} a \,x^{4} b^{2}+\frac {1}{1024} a^{2} x^{4}+\frac {3}{2048} a^{2} x^{5} b\right ] \\ \left [R &= x, S \left (R \right ) &= \int _{}^{y}\frac {1}{\frac {1}{64}+\frac {a^{3} x^{6}}{4096}+\frac {3 a^{2} x^{5} b}{2048}+\frac {3 \left (a \,b^{2}+a^{2} \left (\textit {\_a} +\frac {1}{3}\right )\right ) x^{4}}{1024}+\frac {\left (b^{3}+2 \left (3 \textit {\_a} +1\right ) a b \right ) x^{3}}{512}+\frac {\left (\left (3 \textit {\_a} +1\right ) b^{2}+3 a \left (\textit {\_a} +\frac {2}{3}\right ) \textit {\_a} \right ) x^{2}}{256}+\frac {3 \left (\textit {\_a} +\frac {2}{3}\right ) \textit {\_a} b x}{128}+\frac {\textit {\_a}^{3}}{64}+\frac {\textit {\_a}^{2}}{64}+\frac {b}{128}}d \textit {\_a}\right ] \\ \end{align*}