Answer
$${\text{The function satisfies the given differential equation}}$$
Work Step by Step
$$\eqalign{
& {\text{Let the differential equation be}}\frac{{dy}}{{dx}} = {e^{ - y}} \cr
& {\text{Separate the variables}} \cr
& {e^y}dy = dx \cr
& {\text{Integrate both sides with respect to }}x \cr
& {e^y} = x + C \cr
& {\text{With }}y = 1{\text{ and }}x = 0 \cr
& {e^1} = 0 + C \cr
& C = e \cr
& {\text{then}}{\text{, }} \cr
& {e^y} = x + e \cr
& \cr
& {\text{Solving for }}y \cr
& \ln {e^y} = \ln \left( {x + e} \right) \cr
& y = \ln \left( {x + e} \right) \cr
& {\text{The function satisfies the given differential equation}} \cr} $$