#The purpose of this file is to solve the de eps^2 y''+eps x y' -y+exp(x), y(0) = 2, y(1) = 1
restart;
eq1:=diff(Y0(z),z,z) - Y0(z) + 1;# boundary layer equation at x=0;
ic0:=Y0(0)=2,Y0(infinity)=1;
sol0:=dsolve({eq1,ic0});assign(sol0);
eq2:=diff(Y1(z),z,z) + diff(Y1(z),z) -Y1(z)+exp(1); #boundary layer eqution at y=1;
ic2:=Y1(0)=1,Y1(-infinity)=exp(1);
sol2:=dsolve({eq2,ic2});assign(sol2);
y0 :=exp(x); #outer solution
YY0:=x->subs(z=x/eps,Y0(z)); #left boundary layer
YY1:=x->subs(z=(x-1)/eps,Y1(z)); #right boundary layer
plot(subs(eps=0.01,YY1(x)+YY0(x))+y0-exp(1)-1,x=0..1); #composite solution
JSFH
JSFH