% This code solves x'=f(x)=x  using Euler, 
% second order RK and plots them along
% with the exact solution for comparison.

format compact;
clear all;

% INITIAL CONDITIONS

h = 1;
time = 5;
iter = time / h;

y(1)=1;
z(1)=1;
t(1)=0;

% EULER'S METHOD

for i=1:iter
    y(i+1)=y(i)+h*y(i);
    t(i+1)=i*h;
end

% SECOND ORDER RUNGE_KUTTA METHOD

for i=1:iter
    k1 = h * z(i);
    k2 = h * ( z(i) + k1 );
    z(i+1) = z(i) + h*(k1+k2)/2;
end

% Plot solutions

hold on;
plot(t,z,'b');
plot(t,y,'r');

% Plot exact solution
x = 0:(time/100):time;
plot( x, exp(x),'k');

legend('2-nd order RK','Euler (1-st order)','Exact Solution',3);
hold off;