(************************************************)
(* Singular perturbation: ϵ y'' + y' = (E^(-x));
   y(0) = 1;  y(1) = 1                        *)
(************************************************)

Remove["Global`*"]
(* Exact solution *)
DSolve[{ϵ y''[x]+y'[x]==Exp[-x],y[0]==1,y[1]==1},y[x],x]

Remove["Global`*"]
(*ϵ = 0.1;*)
y[ϵ_]:= (((E^(-1-x)) (-E+2 (E^(1+x))+(E^(1+(1/ϵ)))-(E^(x+(1/ϵ)))-(E^(1+x+(1/ϵ)))+(E^((1/ϵ)+((x (-1+ϵ))/ϵ)))-(E^(1+(1/ϵ)+((x (-1+ϵ))/ϵ)))-(E^(1+x)) ϵ+(E^(1+x+(1/ϵ))) ϵ))/((-1+(E^(1/ϵ))) (-1+ϵ)));
ypert[ϵ_]:= 1-(E^(-x))+(E^(-x/ϵ))-(E^(-(1+x/ϵ)))+(E^(-1));
Plot[{y[0.01],ypert[0.01]},{x,0,1},PlotStyle->Thick]

Remove["Global`*"]
DSolve[y'[x]==Exp[-x],y[x],x]