Link to actual problem [9278] \[ \boxed {y^{\prime }-\frac {-32 x y-8 x^{3}-16 a \,x^{2}-32 x +64 y^{3}+48 y^{2} x^{2}+96 y^{2} a x +12 y x^{4}+48 y a \,x^{3}+48 y a^{2} x^{2}+x^{6}+6 a \,x^{5}+12 a^{2} x^{4}+8 a^{3} x^{3}}{64 y+16 x^{2}+32 a x +64}=0} \]
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 {8 a^{3} x^{3}+12 a^{2} x^{4}+6 x^{5} a +x^{6}+48 a^{2} x^{2} y +48 a \,x^{3} y +12 x^{4} y +96 a x \,y^{2}+48 x^{2} y^{2}+16 x \,a^{2}+8 x^{2} a +64 y^{3}+32 a y +32 a}{32 x a +16 x^{2}+64 y +64}\right ] \\ \left [R &= x, S \left (R \right ) &= \int _{}^{y}\frac {32 x a +16 x^{2}+64 \textit {\_a} +64}{x^{6}+6 x^{5} a +12 \left (a^{2}+\textit {\_a} \right ) x^{4}+8 \left (a^{3}+6 \textit {\_a} a \right ) x^{3}+8 \left (6 \textit {\_a} \,a^{2}+6 \textit {\_a}^{2}+a \right ) x^{2}+16 \left (6 \textit {\_a}^{2} a +a^{2}\right ) x +32 a \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 &= -2 \\ \eta &=x +a \\ \frac {dS}{dR} &= \frac {-R -1}{2 R^{3}+R a +a} \\ \end{align*}