I am interested in finding a clean explicit solution (if possible) to the differential equation
$$ y'(t) = 1-y(t) e^{y(t)-1}, $$ where $0 \le t < 1$ and $0 \le y \le 1$. This can obviously be solved with direct integration, giving the solution
$$ t + C = \int \frac{1}{1-ye^{y-1}} dy. $$
I have not found a solution to the integral, but that would essentially answer my question. I would like an expression for $y(t)$ that does not involve inverse functions. I tried finding a series solution but did not get very far.