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

% The problem to be solved is:

%y'(t)=y-t^2+1
%Note: this problem has a known exact solution
% y(t)= (t+1)^2 - 0.5exp(t)

h=0.5;		%h is the time step.
t=0:h:2;	%initialize time variable.
t1=0:0.01:2;

clear ystar;	%wipe out old variable.

ystar(1)=0.5;	%initial condition (same for approximation).

for i=1:length(t)-1,	%Set up "for" loop.
   k1= ystar(i)-t(i)*t(i)+1.0;  	%Calculate derivative;
   ystar(i+1)=ystar(i)+h*k1;		%Estimate new value of y;
end

%exact solution
y= (t1+1).^2 - 0.5*exp(t1); 

%Plot approximate and exact solution.
plot(t,ystar,'b--',t1,y,'r-',t,ystar,'o');
legend('Approximate','Exact');
title('Euler Approximation to dy/dt=y-t^2+1, h=0.5');
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