I've been struggling to solve equations in two variables containing constant symbolic parameters. Take for example the function in two variables defined by:
F[x_,y_]:=Sqrt[2 t1^2 + t3^2 + 4 t1 t3 Cos[(3 x)/2] Cos[(Sqrt[3] y)/2] + 2 t1^2 Cos[Sqrt[3] y]] My goal is to solve $$F(x,y)=0$$ so I want to find the coordinates of the points in the 2D xy plane where the function is 0. These coordinates will be given by an expression containing the parameters $t_{1}$, $t_{3}$. I can find some solutions by guessing, for example: $$(x,y)=(0,\frac{2}{\sqrt{3}}\arccos{(-\frac{t_{3}}{2t_{1}})}) $$
or $$(x,y)=(0,-\frac{2}{\sqrt{3}}\arccos{(-\frac{t_{3}}{2t_{1}})}) $$
I would like to find these solutions (and the others) in Mathematica,so I've tried:
Solve[F[x, y] == 0, {x, y}] or
Reduce[F[x, y] == 0, {x, y}] However my problem is that Mathematica gives me an expression where the solution for x contains y or viceversa, but I would like the solutions separate (so the solution for x shall not contain y and viceversa, as in the two I wrote above). Also, sometimes the parameters $t_{1}$,$t_{3}$ are treated as variables by Mathematica, and a solution is provided also for them, which I don't want.
This is a problem I had in general for these kind of equations where I solve for 2 variables and I also have additional constant parameters. So how can I get the solution for x and y (without them been coupled) in terms of the constant parameters?
Solve. $\endgroup$SolveuseSolve[F[x, y] == 0, {x, y}, Reals, Method -> Reduce]$\endgroup$