Is there a closed form solution to $e^{-x/b}(a+x) = e^{x/b}(a-x)$? I have the following equation
$$e^{-x/b}(a+x) = e^{x/b}(a-x)$$
where $b > 0$, and $a > 0$
I need to solve for $x$. I can do it numerically, but would prefer if there was a closed form solution.
It seems to me that there likely is no closed form solution, but thought I'd ask the experts here, just in case.

As Steven Taschuk  commented, the equation can write $$\frac xa = \tanh(\frac xb)$$ Changing variable $x=by$, this write $$\frac {b}{a} y = \tanh(y)$$ Taking into account the shape of $\tanh(y)$ and essentially the fact that, at $y=0$, $\Big(tanh(y)\Big)'=1$, there is a solution only if $a\gt b$. Otherwise, the only possible solution is $y=0$ which is valid for any non zero values of $a$ and $b$. If $y_*$ is a solution, $-y_*$ is the other one. So, we have a maximum of three solutions.
Unfortunately, it seems that there is no closed form for the solution and that numerical methods should be required.
Using Taylor expansion to the third order, a first approximation is given by $$y_*= \sqrt{3\frac{a-b}{a}}$$ 
