How can I solve the following equation analytically for t: $t=a(e^t+b)$?

Question

I need to solve $t=a(e^t+b)$ analytically for $t$ knowing that $a$, $b$ are constants, any ideas?

Answer 1

Please see Wikipedia article on Lambert $W$ function.

\begin{align}t = a(e^t+b) &\implies t = ae^t+ab\\ &\implies t -ab = ae^t\\ &\implies (t-ab)e^{ab-t} = ae^{ab}\\ &\implies (ab-t)e^{ab-t} = -ae^{ab}\\ &\implies ab-t = W(-ae^{ab})\\ &\implies t = ab-W(-ae^{ab})\end{align}