Link to actual problem [10375] \[ \boxed {\left (x a +c \right ) y^{\prime }-\alpha \left (a y+b x \right )^{2}-\beta \left (a y+b x \right )=-b x +\gamma } \]
type detected by program
{"riccati", "first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], _rational, _Riccati]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {2 \arctan \left (\frac {2 a^{3} \alpha y+2 a^{2} \alpha x b +a^{2} \beta }{\sqrt {4 a^{4} \alpha \gamma -a^{4} \beta ^{2}+4 a^{3} \alpha b c}}\right )}{\sqrt {4 a^{4} \alpha \gamma -a^{4} \beta ^{2}+4 a^{3} \alpha b c}}\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^{3} \alpha \,y^{2}+2 a^{2} \alpha b x y +a \alpha \,b^{2} x^{2}+a^{2} \beta y +a b \beta x +a \gamma +b c}{b c} \\ \frac {dS}{dR} &= \frac {b c}{a \left (R a +c \right )} \\ \end{align*}