Link to actual problem [8988] \[ \boxed {y^{\prime }+\frac {2 x^{2}+2 x -3 \sqrt {x^{2}+3 y}}{3 x +3}=0} \]
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 &= -\frac {3 x}{2}-\frac {3}{2}, \underline {\hspace {1.25 ex}}\eta &= x^{2}+x\right ] \\ \left [R &= y+\frac {x^{2}}{3}, S \left (R \right ) &= -\frac {2 \ln \left (-x -1\right )}{3}\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 {3 x \sqrt {x^{2}+3 y}+3 \sqrt {x^{2}+3 y}}{2 x +2} \\ \frac {dS}{dR} &= \frac {2}{3 R +3} \\ \end{align*}