Link to actual problem [9213] \[ \boxed {y^{\prime }+\frac {2 a}{-y-2 a -2 y^{4} a +16 a^{2} x y^{2}-32 a^{3} x^{2}-2 y^{6} a +24 y^{4} a^{2} x -96 y^{2} a^{3} x^{2}+128 a^{4} x^{3}}=0} \]
type detected by program
{"first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, _with_linear_symmetries]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \\ \left [R &= y^{2}-4 x a, S \left (R \right ) &= \frac {y}{2 a}\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 &= y \\ \eta &=2 a \\ \frac {dS}{dR} &= -\frac {1}{8 a^{2} \left (R^{3}+R^{2}+1\right )} \\ \end{align*}