Link to actual problem [6195] \[ \boxed {2 x \left (1+\sqrt {x^{2}-y}\right )-\sqrt {x^{2}-y}\, y^{\prime }=0} \]
type detected by program
{"exact", "first_order_ode_lie_symmetry_calculated"}
type detected by Maple
[_exact, [_1st_order, `_with_symmetry_[F(x),G(y)]`]]
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 {1}{\sqrt {x^{2}-y}}\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {2 \left (x^{2}-y\right )^{\frac {3}{2}}}{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 {-2 x^{2} \sqrt {x^{2}-y}+2 \sqrt {x^{2}-y}\, y -3 x^{2}}{\sqrt {x^{2}-y}} \\ \frac {dS}{dR} &= 0 \\ \end{align*}