Expand@DSolve[{x'[t]-x[t]==Exp[-2 t],x[0]==-1/3+ϵ},x[t],t]

Plot[{-(1/3) (E^(-2 t))+(E^t) ϵ/.{ϵ->0},-(1/3) (E^(-2 t))+(E^t) ϵ/.{ϵ->10^-7}},{t,0,18},PlotRange->All,AxesOrigin->{0,-1}]

Plot[Evaluate[x[t]/.First@NDSolve[{x'[t]-x[t]==Exp[-2 t],x[0]==-1/3},x[t],{t,0,18}]],{t,0,18},PlotRange->All]