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

Remove["Global`*"]
(*ϵ = 0.1;*)
y[ϵ_]:= ((((E^(-x))-(E^(-x/ϵ))))/((E^(-1))-(E^((-1)/ϵ))));
(Subscript[y, out])=(E^(1-x));
(Subscript[Y, in])[ϵ_]:=(E^1)(1-(E^(-x/ϵ)));
(Subscript[y, unif])[ϵ_]:=(E^(1-x))-(E^(1-x/ϵ));
N[(E^(-1/0.2))]
Plot[{y[0.01],(Subscript[y, out]),(Subscript[Y, in])[0.01],(Subscript[y, unif])[0.01]},{x,0,1},PlotStyle-> Thick]

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

Remove["Global`*"]
(*ϵ = 0.1;*)
y[ϵ_]:=(((E^(-1-x)) (E-(E^(1+x))-(E^(1+(1/ϵ)))+(E^(x+(1/ϵ)))-(E^(1+x+(1/ϵ)))-(E^((1/ϵ)+((x (-1+ϵ))/ϵ)))+2 (E^(1+(1/ϵ)+((x (-1+ϵ))/ϵ)))+(E^(1+x+(1/ϵ))) ϵ-(E^(1+(1/ϵ)+((x (-1+ϵ))/ϵ))) ϵ))/((-1+(E^(1/ϵ))) (-1+ϵ)));
Plot[{y[0.2],y[0.1],y[0.05],y[0.025]},{x,0,1},PlotStyle-> Thick,PlotRange->{{0,1},{0,2}}]

DSolve[{ϵ y'[x]+y[x]==Sin[x],y[0]==1},y[x],x]

Remove["Global`*"]
(Subscript[y, exact])[ϵ_]:=-(((E^(-(x/ϵ))) (-1-ϵ-(ϵ^2)+(E^(x/ϵ)) ϵ Cos[x]-(E^(x/ϵ)) Sin[x]))/(1+(ϵ^2)));
(Subscript[y, inner])[ϵ_]:=Exp[-x/ϵ]+ϵ ((x/ϵ)-1+Exp[((-x)/ϵ)]);
(Subscript[y, outer])[ϵ_]:=Sin[x];
Plot[{(Subscript[y, exact])[0.1],(Subscript[y, inner])[0.1],(Subscript[y, outer])[0.1]},{x,0,6},PlotStyle->Thick]