Long term behavior of the solution of $u'=e^{-u}-u$

Question

Consider the autonomous differential equation $$\left\{\begin{matrix}u'&=&e^{-u}-u&=:&f(u)\\u(0)&=&u_0&\in&\mathbb{R}\end{matrix}\right.$$ How can we analyze the *long term behavior*$^1$ of its solution $u$?

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$^1\;$ i.e. $\displaystyle\lim_{t\to T_\max}u(t)$

Answer 1

Hint: Phase diagram. Here is the graph of $f$:

$\qquad\qquad\qquad$

And the (very simple) phase diagram associated to this graph:

$\qquad\qquad$

For every $u_0$ the limit $u_\infty$ exists and solves $f(u_\infty)=0$, hence $u_\infty=W(1)\approx.567$.

Answer 2

One little picture says more than a long speech!

Figure below and see :http://mathworld.wolfram.com/LambertW-Function.html

$e^{-u}-u=0$ --> $ u e^u = 1$ --> $u=W(1)$