% Solve Odes using Euler method
%  Einat 

function doall=euler
clear all; close all;



x(1)=1;  %initial condition, x(t0)
t(1)=0;  %is taken at t0=0
tf=1;    %final point 
x2(1)=x(1);
xt(1)=x(1);
xanalytical(1)=x(1);

for d=0:4
  dt=10^(-d); %time-step size
  N=tf/dt; %number of time steps

  %1st and 2nd order euler method

  for n=1:N
    x(n+1)=x(n)+dt*rhs(x,n); %1st order
    xt(n+1)=x2(n)+dt*rhs(x2,n); %trail of 2nd order
    x2(n+1)=x2(n)+0.5*dt*(rhs(x2,n)+rhs(xt,n+1)); %2nd order
    t(n+1)=t(n)+dt;
    xanalytical(n+1)=exp(-t(n+1));
  end

  figure
  plot(t,x,'x-')
  hold on
  plot(t,x2,'ro-')
  plot(t,xanalytical,'g')
  title(dt)
  xlabel('t')
  ylabel('x(t)')
  legend('1st order','2nd order','analytical')
end





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% return f(n):
function fn=rhs(x,n)

fn=-x(n);