% apple_quiet.m  :  a variant of apple.m that has semicolons (;) so that
%   the program runs quietly

% A program to solve y' = 2y, and y(t_0)=y_0. 
% What is is really doing is computing (1+2h/N)^N y_0, i.e.,
%     i.e., (1+2(t_end-t_0)/N)^N y_0
% Try running:
%
% 
% sol_three = apple_quiet(1,2,1000,3);
% sol_three(1)     % this should be t_0
%
% sol_three(1001)  % this should equal the approximation to y(t_end), which
                   % due to the simple nature of f, turns out to be exactly
                   %(1+2(t_end-t_0)/N)^N y_0
% sol_three(1001), (1+2*(2-1)/1000)^1000 * 3

% sol_four = apple_quiet(1,2,10000,3);
% sol_four(10001)

% sol_five = apple_quiet(1,2,100000,3);
% sol_five(100001)
% (1+2*(2-1)/100000)^100000 * 3             % This should equal the last line
% exp( 2 * (2-1) ) * 3                      % The exact solution to the ODE

function result = orange(t_0,t_end,N,y_0)  % the name "orange" is irrelevant
                               % MATLAB calls this function "apple_quiet"
 h = (t_end - t_0)/N;
 t = [t_0 : h: t_end];
  y(1) = y_0;
  for i=1:N
    y(i+1) = y(i) + h * f(t(i),y(i));
  end
  result = y;
end

% Second function: this is where we specify the function f.
% This function is purely local to this program.

function two_y = f(t,y)
  two_y = 2 * y;
end