Solving equation for x, with Lambert function

Question

today I was trying to solve an equation for the movement of a particle with air resistance, but I was unable to solve for the falling time. The equation that I need to solve is of this type $0=k+x+nxe^{(nx)} $, n and k constants , I think it is necessary to use Lambert function but I don't now how to get rid of the second term

Answer 1

Rewrite the equation $$e^{nx}=-\frac{x+k}{nx}\implies e^{-nx}=-\frac{nx}{x+k}$$ and the solution (if any) would be given in terms of the generalized Lambert function (have a look at equation $(4)$).

I must confess that this would not be very easy and I suppose that you will need numerical methods for finding the zero(s) of function $$f(x)=k+x+nxe^{nx}$$