function   myeuler(n,xs,ys,xe,s)

%  or you could use    
%     function [x,y]=   myeuler(n,xs,ys,xe,s)
%  if you want the  x,y values as output of program

%  general format for a function file is 
%    function [list of output variables] = Name(list of input variables)
%      your instructions or body of program
%    end

%  Comments are preceded by a percent sign
%  This program is a simple Euler solver for the first order ODE y'=f(x,y)
%  Sample RUN command for this file is:
%    you probably want numerical output then use (inclusive of semi-colon)
%      myeuler(1000,1,3,10,'y'); 
%    if you don't want numerical output use
%      myeuler(1000,1,3,10,'n');
%  you need to understand the RUN command to use it for homework 

%  Explanation of variables:
%  (xs,ys)   is the starting point, the initial condition

%  xe        is the ending x value, solution is found on interval [xs,xe]
%  n         is the number of intervals, total number of points is n+1
%               points are (x(1),y(1)) through (x(n+1),y(n+1))      
%  s         is a  character string, 
%                if you want numerical ouput enter 'y',  otherwise enter 'n', with quotes
%  f         is a function given in the subroutine below. edit it for your ODE

h=(xe-xs)/n;        % also named: run, increment, delta, delta x, or step-size
 
x=zeros(1,n);       % allocating space for a vector for x values

y=zeros(1,n);       % allocating space for a vector for y values

x(1)=xs;            % initialize the first x value

y(1)=ys;            % initialize the first y value


   for i=1:n        % start a "for" loop, i runs from 1 to n, 
                      % the generated points here are x(2),...,x(n+1)   
     x(i+1)=x(i)+h;   % increase x by the run/step-size/delta/delta x
  
     m=f(x(i),y(i));  % m is slope, a call is made to function f, below, to calculate m  
     k=h*m;           % k is rise=run*slope
   
     y(i+1)=y(i)+k;   % to get new y, increase old y by rise
   end              % end of  "for" block, program repeats this block n times

plot(x,y)           % plot the solution, y is on vertical axis, x is on horizontal axis

   if s=='y'        % start an "if" block, if the string s has value 'y' then
     [x;y]'           % program will print out [x;y] values as a vertical table
                      % Note: if v is a horizontal vector then v' is a column vector
   end              % end of "if" block, if the string s has any other value nothing is printed

end                 % end of the main function, end of myeuler


function yp=f(x,y)  % this subroutine calculates slope in terms of position 

                    %   f is the right hand side of ODE, as in dfield software   
                    %   this has to be edited for each problem 
yp=sin(5*x*y);      %   this particular example is for the ODE  y'=sin(5xy). 
                    % Note: you need a * to indicate multiplication
                    % Note: be consistent with name of variables 
                    %      if function is f(x,y) then use x,y inside this subrotine
                    %      if function is f(t,y) then use t,y inside
                    % Edit and Save this formula for your ODE
end                 % end of subroutine