What is the solution for $y(t)=e^{-\frac{t}{\tau y(t)}}$?

Question

A simple quadratic flow model leads to the following apparently simple equation

$$y(t)=e^{-\frac{t}{\tau y(t)}}$$

where the flow, $y$ is a function of time, $t$ and $\tau $ is a constant.

But is there a closed form solution for $y(t)$ just in terms of $t$ and $\tau$ ?

Or can this only be solved by numerical means ?

Answer 1

There is a "closed-form" expression in terms of the Lambert W Function.

We begin with the equation

$$y(t)=e^{-t/\tau y(t)} \tag 1$$

Let $z=-t/\tau$ and let $W=\frac{z}{y(t)}$. Then, upon rearranging $(1)$ we find that

$$z=We^W \tag 2$$

Noting that $(2)$ defines the Lambert W, we have immediately that

$$\bbox[5px,border:2px solid #C0A000]{y(t)-\frac{-t/\tau}{W\left(-t/\tau\right)}}$$

Special attention is required to determine the domain for which $y(t)$ is real and single valued.