% Demonstration of  Euler's method for IVP x'=x, x(0)=1.
% Cut and paste this whole file into the Matlab command window
% or else download it and invoke it as "eulerdemo"
% MVW, 2007-11-16
%
n=10;        % Number of steps
x(1)=1;      % Initial value
h=1/n;       % Compute step size
t=h*(0:n);   % Compute sample times    

% Generate vector of approximate samples by Euler's method:
for k=2:n+1,
   x(k)=x(k-1)+h*x(k-1);
end

% Generate exact solution:
for k=0:n,
   y(k+1)=exp(k/n);
end

% Show differences:
figure; plot(t,x,'red',t,y,'blue');  % Euler and exact on the same axes
figure; plot(t,y-x);                 % Difference exact-Euler