Link to actual problem [9108] \[ \boxed {y^{\prime }-\frac {-4 y x -x^{3}-2 a \,x^{2}-4 x +8}{8 y+2 x^{2}+4 x a +8}=0} \]
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 {2 x \,a^{2}+x^{2} a +4 a y +4 a +8}{4 x a +2 x^{2}+8 y +8}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {2 y}{a}-\frac {4 \ln \left (2 x \,a^{2}+x^{2} a +4 a y+4 a +8\right )}{a^{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 {2 x \,a^{2}+x^{2} a +4 a y +4 a +8}{2 x a +x^{2}+4 y +4} \\ \frac {dS}{dR} &= -\frac {R}{2 a} \\ \end{align*}