%% Improved Euler method %% 
%% Written for x'=f(x) with f(x)=-x and x(0)=x0
clear all

dt=.1;
x0=0;
t0=0;

endtime=1;

t=0:dt:endtime;

N=(endtime-t0)/dt;

x(1)=x0;
for n=1:N

    x1=x(n)+dt*(x(n)+exp(-x(n)));
    
    x(n+1)=x(n) + 0.5*dt*( (x(n)+exp(-x(n))) + (x1+exp(-x1)) )
    
end

plot(t,x,'k')
hold on


%% END IMPROVED EULER METHOD %%