Solve $t\, e^t=t+2$ for $t>0$. I need to solve the following equation:
$$t\, e^t=t+2\, ,$$
for $t>0$. Do I need to use Lambert W function, or there is some other method?
Thanks!

PS. I know that any equation of the form $A+Bt+C\log(D+Et)=0$ has solutions which, if they exist, can be expressed using Lambert function. But I do not think that $t\, e^t=t+2$ can be rewrite in this way... am I wrong?


I do not believe that this can be solved using the Lambert $W$ function. When one attempts to put it in the form that you specified, one ends up with
$$\ln\bigg(\frac{t}{t+2}\bigg)+t=0$$
If we let $\frac{t}{t+2}=u$, then we have
$$\ln(u)+\frac{2u}{1-u}=0$$
$$(1-u)\ln(u)+2u=0$$
So if you can find some way of solving equations in the form
$$A+Bt+(C+Dt)\ln(E+Ft)=0$$
then this would be possible... however, I am afraid that you may have to attack this problem without the Lambert $W$.
If you would like to approximate, I would recommend using iteration.
