Link to actual problem [8574] \[ \boxed {\left (x \left (y+x \right )+a \right ) y^{\prime }-y \left (y+x \right )=b} \]
type detected by program
{"first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, `2nd type`, `class B`]]
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 (a y -b x \right ) \left (x^{2} a +2 a x y -b \,x^{2}+a^{2}\right )}{x^{2}+x y +a}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {\ln \left (x^{2} a +2 a x y-b \,x^{2}+a^{2}\right )}{2 a^{2}}+\frac {\ln \left (a y-b x \right )}{a^{2}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\left (a y -b x \right ) \left (2 a^{2} y^{2}+3 a b \,x^{2}+2 a b x y -b^{2} x^{2}+3 a^{2} b \right )}{x^{2}+x y +a}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {\ln \left (2 y^{2} a^{2}+3 a b \,x^{2}+2 a b x y-b^{2} x^{2}+3 a^{2} b \right )}{6 a^{2} b}+\frac {\ln \left (a y-b x \right )}{3 a^{2} b}\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 \,x^{2} y -2 a x \,y^{2}-a \,y^{3}+b \,x^{3}+2 b \,x^{2} y +b x \,y^{2}-a^{2} y +a b x -a b y +b^{2} x}{x^{2} a +a x y +a^{2}} \\ \frac {dS}{dR} &= 0 \\ \end{align*}