$e^{-xy} + e^{xy} = 2e^{-y}$ - where am I going wrong? I am trying to see if there is any $x$ (real or complex) for which this equation can be solved.
$$e^{-xy} + e^{xy} = 2e^{-y}$$
Step 1. Multiplying both sides by y,
$$ye^{-xy} + ye^{xy} = 2ye^{-y}$$
Step2. Partially differentiating the original equation (i.e. $e^{-xy} + e^{xy} = 2e^{-y}$ ) w.r.t. x,
$$-ye^{-xy} + ye^{xy} = 0$$
Step3. Adding Steps 1 and 2,
$$2ye^{xy} = 2ye^{-y}$$
this gives $x=-1$.
Obviously this is not a solution. Where am I going wrong? or is there a complex value of x for which Step 3 holds?

The derivative of two different constant functions is zero.  The two functions are not equal.  All equality of derivatives tells you is that the two functions agree up to a constant offset (recall the "${}+C$" from integral calculus).
The partial derivative with respect to $x$ of two different functions depending only on $y$ is zero.  All equality of partial derivatives with respect to $x$ tells you is that the two functions agree up to an arbitrary function in $y$.  Your right-hand side is not "$0$"; it's "$F(y)$", which isn't nearly as convenient for your solution method.
