3D representation of a transcendental equation

Question

Given a transcendental equation $F * exp [x] + x * y + sqrt[F-y] = 0$, is there a way for Mathematica to automatically 3D-plot F[x,y] depending on x and y ?

Answer 1

The simplest method would be just

ContourPlot3D[F Exp[x] + x y + Sqrt[F - y] == 0, {x, -2, 2}, {y, -2, 2}, {F, -2, 2}]

Unfortunately there are some problems because the Sqrt term is complex sometimes. A better plot is obtained by squaring the equation and using RegionFunction to eliminate the extra solutions:

ContourPlot3D[(F Exp[x] + x y)^2 == F - y, {x, -2, 2}, {y, -2, 2}, {F, -2, 2}, 
 RegionFunction -> Function[{x, y, F}, F Exp[x] + x y < 0]]

Answer 2

Not a great deal. First you solve it:

    sl = Solve[F*Exp[x] + x*y + Sqrt[F - y] == 0, F]

(*  {{F -> 1/2 E^(-2 x) (1 - 2 E^x x y - Sqrt[
      1 - 4 E^(2 x) y - 4 E^x x y])}, {F -> 
   1/2 E^(-2 x) (1 - 2 E^x x y + Sqrt[1 - 4 E^(2 x) y - 4 E^x x y])}}  *)

and then plot:

 Plot3D[{sl[[1, 1, 2]], sl[[2, 1, 2]]}, {x, -1, 1}, {y, -1, 1}, 
 PlotStyle -> {Blue, Orange}]

and should see this:

Have fun!