Link to actual problem [8996] \[ \boxed {y^{\prime }-x^{2} \sqrt {x^{2}+2 x +1-4 y}=\frac {x}{2}+\frac {1}{2}} \]
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 [\underline {\hspace {1.25 ex}}\xi &= -x, \underline {\hspace {1.25 ex}}\eta &= x^{2}+\frac {5}{2} x -6 y +\frac {3}{2}\right ] \\ \left [R &= -\frac {x^{2}+2 x +1-4 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}+2 x -4 y +1}+\frac {3 x^{2}}{2}+3 x -6 y +\frac {3}{2} \\ \frac {dS}{dR} &= 0 \\ \end{align*}