Link to actual problem [8585] \[ \boxed {\left (y a x +b \,x^{n}\right ) y^{\prime }+\alpha y^{3}+\beta y^{2}=0} \]
type detected by program
{"first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`], [_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 &= \frac {x \left (\alpha y +\beta \right )}{\alpha }, \underline {\hspace {1.25 ex}}\eta &= \frac {y \left (\alpha y +\beta \right ) \left (a n -\alpha y -a \right )}{a \alpha }\right ] \\ \left [R &= \frac {\left (a n -\alpha y-a \right ) x^{n -1}}{a y \left (n -1\right )}, S \left (R \right ) &= -\frac {\alpha \left (a \ln \left (\alpha y+\beta \right ) n -\ln \left (y\right ) a n +\ln \left (-a n +\alpha y+a \right ) \beta -a \ln \left (\alpha y+\beta \right )+a \ln \left (y\right )-\ln \left (y\right ) \beta \right )}{\left (n -1\right ) \left (a n -a +\beta \right ) \beta }\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 &= 0 \\ \eta &=\frac {a^{2} \alpha n x \,y^{3}+x^{n} a \alpha b n \,y^{2}-x^{n} \alpha ^{2} b \,y^{3}-a^{2} \alpha x \,y^{3}+a^{2} \beta n x \,y^{2}+a \alpha \beta x \,y^{3}-x^{n} a \alpha b \,y^{2}+y \,x^{n} a b \beta n -x^{n} \alpha b \beta \,y^{2}-a^{2} \beta x \,y^{2}+a \,\beta ^{2} x \,y^{2}-y \,x^{n} a b \beta }{a^{2} \alpha y x +a \alpha b \,x^{n}} \\ \frac {dS}{dR} &= -\frac {\alpha n}{R \left (n -1\right ) \left (\left (n -1\right ) a +\beta \right )} \\ \end{align*}