Link to actual problem [3814] \[ \boxed {x \left (x^{n}+a y\right ) y^{\prime }+\left (b +c y\right ) y^{2}=0} \]
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 {a x \left (c y +b \right )}{c^{2}}, \underline {\hspace {1.25 ex}}\eta &= -\frac {y \left (c y +b \right ) \left (a n -c y \right )}{c^{2}}\right ] \\ \left [R &= \frac {\left (a n -c y\right ) x^{n}}{a n y}, S \left (R \right ) &= \frac {c^{2} \left (\ln \left (b +c y\right ) a n -\ln \left (y\right ) a n +\ln \left (-a n +c y\right ) b -\ln \left (y\right ) b \right )}{b a n \left (a n +b \right )}\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} c n \,y^{3}-x^{n} a c n \,y^{2}+x^{n} c^{2} y^{3}-a^{2} b n \,y^{2}-a b c \,y^{3}-y \,x^{n} a b n +x^{n} b c \,y^{2}-a \,b^{2} y^{2}}{c^{2} x^{n}+c^{2} a y} \\ \frac {dS}{dR} &= \frac {c^{2}}{R \left (a n +b \right ) a} \\ \end{align*}