Link to actual problem [9277] \[ \boxed {y^{\prime }-\frac {-32 a x y-8 a^{2} x^{3}-16 a b \,x^{2}-32 a x +64 y^{3}+48 y^{2} a \,x^{2}+96 y^{2} b x +12 y a^{2} x^{4}+48 y a \,x^{3} b +48 y b^{2} x^{2}+a^{3} x^{6}+6 a^{2} x^{5} b +12 a \,b^{2} x^{4}+8 b^{3} x^{3}}{64 y+16 a \,x^{2}+32 x b +64}=0} \]
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 {a^{3} x^{6}+6 a^{2} x^{5} b +12 a^{2} x^{4} y +12 a \,x^{4} b^{2}+48 a b \,x^{3} y +8 b^{3} x^{3}+48 a \,x^{2} y^{2}+48 b^{2} x^{2} y +8 a b \,x^{2}+96 b x \,y^{2}+16 b^{2} x +64 y^{3}+32 b y +32 b}{1024 x^{2} a +2048 b x +4096 y +4096}\right ] \\ \left [R &= x, S \left (R \right ) &= \int _{}^{y}\frac {1024 x^{2} a +2048 b x +4096 \textit {\_a} +4096}{a^{3} x^{6}+6 a^{2} x^{5} b +12 \left (\textit {\_a} \,a^{2}+a \,b^{2}\right ) x^{4}+8 \left (6 \textit {\_a} a b +b^{3}\right ) x^{3}+8 \left (6 \textit {\_a}^{2} a +6 \textit {\_a} \,b^{2}+a b \right ) x^{2}+16 \left (6 \textit {\_a}^{2} b +b^{2}\right ) x +32 b \left (\textit {\_a} +1\right )+64 \textit {\_a}^{3}}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 &= -\frac {2}{b} \\ \eta &=\frac {x a +b}{b} \\ \frac {dS}{dR} &= -\frac {\left (R +1\right ) b}{b \left (R +1\right )+2 R^{3}} \\ \end{align*}