Link to actual problem [8627] \[ \boxed {\left (b \left (\beta y+x \alpha \right )^{2}-\beta \left (x a +b y\right )\right ) y^{\prime }+a \left (\beta y+x \alpha \right )^{2}-\alpha \left (x a +b y\right )=0} \]
type detected by program
{"first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], _rational]
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 {\left (\alpha x +\beta y \right )^{2} \left (x a +b y \right )}{-\alpha ^{2} b \,x^{2}-2 \alpha b \beta x y -b \,\beta ^{2} y^{2}+a \beta x +b \beta y}\right ] \\ \left [R &= x, S \left (R \right ) &= -\ln \left (x a +b y\right )-\frac {1}{\beta y+\alpha x}\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 ^{2} \beta \,x^{3}-2 a^{2} \alpha \,\beta ^{2} x^{2} y -a^{2} \beta ^{3} x \,y^{2}+a \,\alpha ^{3} b \,x^{3}+a \,\alpha ^{2} b \beta \,x^{2} y -a \alpha b \,\beta ^{2} x \,y^{2}-a b \,\beta ^{3} y^{3}+\alpha ^{3} b^{2} x^{2} y +2 \alpha ^{2} b^{2} \beta x \,y^{2}+\alpha \,b^{2} \beta ^{2} y^{3}}{-\alpha ^{2} b^{2} \beta \,x^{2}-2 \alpha \,b^{2} \beta ^{2} x y -b^{2} \beta ^{3} y^{2}+a b \,\beta ^{2} x +b^{2} \beta ^{2} y} \\ \frac {dS}{dR} &= 0 \\ \end{align*}