I want to solve system of equations \begin{cases} 2 x^2-5 x y-y^2=y \left(\sqrt{x y-2 y^2}+\sqrt{4 y^2-x y}\right),\\ \sqrt{3 y}+\sqrt{x^2+2 x}-x-x \sqrt{2+9 y^2}=0. \end{cases} Where $x$, $y$ are two real numbers. I tried
Reduce[{2 x^2 - 5 x y - y^2 ==
y (Sqrt[x y - 2 y^2] + Sqrt[4 y^2 - x y]),
Sqrt[3 y] + Sqrt[x^2 + 2 x] - x - x Sqrt[2 + 9 y^2] == 0}, {x,
y}, Reals]
I can't get solutions for a long time. I know that, the given system of equations has two solutions $(0,0)$ and $(1,1/3)$. If I use NSolve
NSolve[{2 x^2 - 5 x y - y^2 ==
y (Sqrt[x y - 2 y^2] + Sqrt[4 y^2 - x y]),
Sqrt[3 y] + Sqrt[x^2 + 2 x] - x - x Sqrt[2 + 9 y^2] == 0}, {x,
y}, Reals]
I only got
{{y -> 0.333333, x -> 1.}, {y -> 0.333333, x -> 1.}}
Lost solution $(0,0)$.
How can I get exact solutions of this system of equations?
Reduce::ivar
error. $\endgroup$Reduce
should receive a single equation or a list of equations as its first argument, so you should wrap your equations in{}
. $\endgroup$GroebnerBasis[]
as a preprocessor before feeding toSolve[]
. $\endgroup$NSolve
gives me an empty solution set, whereasFindInstance
gives me only $\left( 0,0\right)$. $\endgroup$