Finding a Fixed Point Solution

Question

I'm trying to solve the following where $A,B >0$ in the most general case (don't assume they are equal). Wolfram Alpha cannot compute this in the allowed time, but I feel some fixed point algorithm should work. I'm not too familiar with numerics, could someone find a solution via Matlab etc...

$$\frac{1}{\pi}\arctan\left(\frac{(B-A)}{(\sqrt{x}+\frac{(A*B)}{\sqrt{x}})}\right)=\sqrt{x}$$

Answer 1

I'm not surprised that WolframAlpha cannot solve the equation directly. It's a difficult equation that Mathematica's Reduce command can't deal with. However, it's easy to solve given specific values of $A$ and $B$. In wolfram alpha, you can just type the input directly with specific values of $A$ and $B$ to solve or plot. For example, here's how to find the solution for $A=0.2$ and $B=1$:

If you have Mathematica, you can define a function of $A$ and $B$ that returns the solution and then visualize the function via a plot or contour plot. Here are the results:

The function definition is

f[A_?NumericQ, B_?NumericQ] := x /. FindRoot[
  ArcTan[(B - A)/((A*B)/Sqrt[x] + Sqrt[x])]/Pi == Sqrt[x],
  {x, 0.1, 0.0, 0.2}]