Answer
$${\text{The function satisfies the given differential equation}}$$
Work Step by Step
$$\eqalign{
& {\text{Let the differential equation be}}\frac{{dy}}{{dx}} = {e^{ - y}} \cr
& {\text{Separate the variables}} \cr
& {e^{ - y}}dy = dx \cr
& {\text{Integrate both sides with respect to }}x \cr
& - {e^{ - y}} = x + C \cr
& {\text{With }}y = - 2{\text{ and }}x = 0 \cr
& - {e^{ - \left( { - 2} \right)}} = 0 + C \cr
& C = - {e^2} \cr
& {\text{then}}{\text{, }} \cr
& - {e^{ - y}} = x - {e^2} \cr
& {e^{ - y}} = {e^2} - x \cr
& {\text{Solving for }}y \cr
& \ln {e^{ - y}} = \ln \left( {{e^2} - x} \right) \cr
& y = - \ln \left( {{e^2} - x} \right) \cr
& {\text{The function satisfies the given differential equation}} \cr} $$