Link to actual problem [9105] \[ \boxed {y^{\prime }-\frac {-4 a x y-a^{2} x^{3}-2 a b \,x^{2}-4 x a +8}{8 y+2 a \,x^{2}+4 x b +8}=0} \]
type detected by program
{"first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, `2nd type`, `class A`]]
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 {a b \,x^{2}+2 b^{2} x +4 b y +4 b +8}{2 x^{2} a +4 b x +8 y +8}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {2 y}{b}-\frac {4 \ln \left (a b \,x^{2}+2 b^{2} x +4 b y+4 b +8\right )}{b^{2}}\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 b \,x^{2}+2 b^{2} x +4 b y +4 b +8}{a^{2} x^{2}+2 a b x +4 a y +4 a} \\ \frac {dS}{dR} &= -\frac {a^{2} R}{2 b} \\ \end{align*}