Answer
$$ y= e^{1-e^{-t}}.$$
Work Step by Step
By separation of variables, we have
$$\frac{dy}{y}=e^{-t}dt $$
then by integration, we get
$$\ln y=-e^{-t} +c\Longrightarrow y= e^{c-e^{-t}}.$$
Now, since $y(0)=1$, then $c=1$.
So the general solution is given by $$ y= e^{1-e^{-t}}.$$
