%Script that demonstrates Euler integration for a first order problem using Matlab.

% The problem to be solved is:

%y'(t)+2*y(t)=3*exp(-4*t)

%Note: this problem has a known exact solution
% y(t)=2.5*exp(-2*t)-1.5*exp(-4*t)

h=0.1;		%h is the time step.
t=0:h:4;	%initialize time variable.

clear ystar;	%wipe out old variable.

ystar(1)=1.0;	%initial condition (same for approximation).

for i=1:length(t)-1,	%Set up "for" loop.
   k1=3*exp(-4*t(i))-2*ystar(i);	%Calculate derivative;
   ystar(i+1)=ystar(i)+h*k1;		%Estimate new value of y;
end

%exact solution
y=2.5*exp(-2*t)-1.5*exp(-4*t);

%Plot approximate and exact solution.
plot(t,ystar,'b--',t,y,'r-');
legend('Approximate','Exact');
title('Euler Approximation, h=0.01');
xlabel('Time');
ylabel('y*(t), y(t)');

%Print results
for i=1:length(t)
   disp(sprintf('t=%5.3f,  y(t)=%6.4f,  y*(t)=%6.4f',t(i),y(i),ystar(i)));
end