% apple.m  :  a program to solve y' = 2y, and y(t_0)=y_0,
% from time t_0 to time t_end.
% We use N steps of Euler's method.

% Here are some points to consider:
% (1) If you don't type a semicolon (;) at the end of a line, you will
%    print the contents of the MATLAB computation to your Command Window
%    (or your terminal if you are running from your Mac OS terminal window:
%    "/Applications/MATLAB_R2023b.app/bin/matlab -nodesktop -nosplash"
%
% (2) Note the function name "orange" -- when you type:
%        apple(1,2,10,3),
%    MATLAB will look in the file apple.m, and assume that the first
%    function defined is the function you wish to run.  Any other
%    functions defined elsewhere are only local functions, and won't
%    run globally...

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
% HOMEWORK: type the following commands into the command window (without the %'s).
%
% sol_one = apple(1,2,10,3)
% size(sol_one)
% sol_one(11)
%
% sol_two = apple(1,2,100,3)
% size(sol_two)
% sol_two(101)
%
% sol_three = apple(1,2,1000,3)
% sol_three(1001)
%
% exp( 2 * (2-1) ) * 3 
%
% sol_one(11),sol_two(101),sol_three(1001),exp( 2 * (2-1) ) * 3
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Here is the code that will be run:

% First function: solves y'=f(t,y) subject to y(t_0)=y_0
% using Euler's method, from t = t_0  to  t = t_end, using 
% step size  h = (t_end-t_0)/N
%

function result = orange(t_0,t_end,N,y_0)  % the name "orange" is irrelevant
                                           % MATLAB calls this function "apple"
 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