clear       % good to start by clearing out all old values
dt=0.001    % dt is the time step
t=0:dt:10;  % t is a vector of numbers from 0 to 10 in steps of dt
x=cos(t);   % x is a vector the same size as t, with each element the cosine of the same element in t
y(1)=0.0;   % redundant because next line sets all values of y to zero
y=zeros(size(t));   % y is a vector the same size as t with all elements set to zero
for i=2:length(t)   % beginning of a loop, that starts with i=2, then i=3, then i=4 until i=no. of elements in t
    y(i) = y(i-1) + dt*x(i-1); % change in y is dt*x(i-1) solves the eqn: dy/dt = x
end             % this end marks the end of the for loop, so go back to line 7 and increase i unless i=length(t) already
subplot(2,1,1); % plot a figure with 2 rows and one column of graphs: start at position 1.
plot(t,x);      % plot x against t
subplot(2,1,2); % go to graph in the second position of the 2x1 set of graphs
plot(t,y);      % plot y against t
hold on;        % don't erase this plot so we can get two curves on one graph
for i=2:length(t)    %loop in the same way as before
    y(i) = y(i-1) + dt*x(i-1) + sqrt(dt)*2*(rand-0.5); % generate y as before but with an extra noise term 
end             % go back to line 15 to increase i, unless i has reached length(t)
plot(t,y,'r');  % plot the new set of values of y against t, in red.