Is there a closed form solution to $e^{-x/b}(a+x) = e^{x/b}(a-x)$?

Question

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.

Answer 1

Is there a closed form solution to this equation ?

No. Not even one in terms of Lambert's W function.

Answer 2

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 $$y_*= \sqrt{3\frac{a-b}{a}}$$

Answer 3

Simple rearranging of the equation shows that the main theorem of [Lin 1983] can be applied that implies that the equation doesn't have solutions that are elementary numbers.

Simple rearranging of the equation gives

$$\frac{2a-bz}{2a+bz}e^z=1$$

This shows that the equation is in a form that is not solvable in terms of Lambert W. But the equation can be solved by Generalized Lambert W:

$$z=W(^{+\frac{2a}{b}}_{-\frac{2a}{b}};-1)$$

$-$ see the references below.
$\ $

[Lin 1983] Ferng-Ching Lin: Schanuel's Conjecture Implies Ritt's Conjectures. Chin. J. Math. 11 (1983) (1) 41-50

[Mezö 2017] Mezö, I.: On the structure of the solution set of a generalized Euler-Lambert equation. J. Math. Anal. Appl. 455 (2017) (1) 538-553

[Mezö/Baricz 2017] Mezö, I.; Baricz, A.: On the generalization of the Lambert W function. Transact. Amer. Math. Soc. 369 (2017) (11) 7917–7934 (On the generalization of the Lambert W function with applications in theoretical physics. 2015)

[Castle 2018] Castle, P.: Taylor series for generalized Lambert W functions. 2018

Answer 4

Let’s see if the equation can be solved with the Inverse of the Regularized Gamma function. If I cannot, or others, solve it with the function, then there is probably no closed form. Here is a similar set up with the Generalized Regularized Gamma function:

$$Q\left(2,\frac xb,-\frac xb\right)=0\iff e^\frac xb\left(\frac xb-1\right)+e^{-\frac xb}\left(\frac xb+1\right)=0$$

Where the inverse cannot be taken with the Inverse of the Regularized Gamma function proving that one cannot use this function, which is a more general Lambert W function.

Let me try to find a series reversion of the solution soon. Alternatively, you will have a closed form with the Root function.