Solution for $ y(x) - ae^{y(x)} = f(x) $ involving Lambert W function

Question

I need to solve an equation of the type:

$$y(x) -ae^{y(x)} = f(x)$$

with $a>0$. Furthermore, the expression for $f(x)$ can't be evaluated analytically (it's the solution of a differential equation that I find applying finite differences).

I read about the Lambert W function and I was wondering if one could apply it to find a general solution of this type of equation. Here it's given a solution for a similar case involving real constants, but I am not sure if the same result could be applied in the case where one has a function $f(x)$.

Answer 1

@Yuriy S made nice comments to which I should not add anything.

The solution is "simply" $$y(x)= f(x)-W\left(-a\, e^{f(x)}\right)$$ The problem is now the fact that Lambert function is multivalued and, depending on $a$ and on the value of $f(x)$ for a given $x$, you could be obliged to change from one branch to another one.