Link to actual problem [8998] \[ \boxed {y^{\prime }-x^{2} \sqrt {x^{2}+4 y-4 x}=1-\frac {x}{2}} \]
type detected by program
{"first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[[_1st_order, `_with_symmetry_[F(x),G(x)]`]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\).\begin{align*} \\ \left [R &= \frac {x^{2}-4 x +4 y}{4 x^{6}}, S \left (R \right ) &= \ln \left (x \right )\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 &=-x^{3} \sqrt {x^{2}-4 x +4 y}+\frac {3 x^{2}}{2}-6 x +6 y \\ \frac {dS}{dR} &= 0 \\ \end{align*}