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%%%%%  The code below concerns the ODE:    x' = 3 sin x   with  x(2) = 6. %%
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% Minimalist solution of ODE with  plot using MATLAB: 2 commands

[t  z] =  ode23('myfunction',  2,   5,   6);
plot( t, z ) 

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% Put the two lines below in  a  file called   myfunction.m before
% typing the two commands above.
% The first line of the file has to start with the word   function.

function    zdot = myfunction( t ,  z )
zdot =  3 * sin ( z );


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%%%%%%%   Use stuff above (to top)  OR  stuff below (to bottom), not both.%%
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% More involved solution of a first order differential equation
% and a plot of the solution    -    using MATLAB.
% By Andy Ruina   (8/30/96)

% (This solves the same problem as the minimalist code above
%   but with more comments,  more accuracy, 
%   more flexibility, and a nicer looking plot.
%     ---  But it looks scarier at first glance.)

% The commands in this part of the solution (down to the line of %s)
% could be enterred as individual commands or, probably better,
% put together in a script file.

global  C  % This allows information to be passed to a function
           % without a direct pass.
           % It is convenient here because the MATLAB function
           % ODE23 does not include provision for passing 
           % parameter values to the 'function' which defines
           % the right hand side of the differential equation.

C = 3;     % assign a value to the constant C

tzero  = 2; % define the start value of t as 2
ffinal = 5; %            final value of t as 5
zzero  = 6; % value of z at tzero is 6, ie.,  z(tzero) = 6

tol = 1e-8; % tol is an optional argument of ODE23 that sets
            % the accuracy of the solution.  
            % The default is 1e-6.   Smaller values of
            % tol make the solution more accurate and slower
            % to compute.  Actually, the solutions get less
            % accurate (because of numerical roundoff) if
            % tol is too small.

            % One way of working is to use tol as something
            % big (like  tol = 1e-3) while debugging your work
            % to make the execution time less.
            % Then set tol to something smaller once all
            % your code works.

% the actual solution is done in this one line that follows these
% comments.
% The result of this command is two arrays:  t  and z, where
% t contains a list of times (chosen by the computer but with
% the first element tzero and the last tfinal) and z an array
% of values of z at those times.
% Note that ode23 is not capitalized even though it is 
% capitalized in MATLAB documentation (such as if you
% type   >help ode23   at the MATLAB prompt.

[t z] = ode23 ('myfunction', tzero, tfinal, zzero, tol );  % the key line


plot( t, z) ; % the plot command in its simplest form is this

xlabel('time') % this adds the text 'time' under the x axis
ylabel('z')    % similar to above for z axis
title('Andys plot of an ODE solution:  8/30/96') % adds a title to the plot
          % Note you cannot use an apostrophe in the title as that
          % is the quote character used to dilineate the text.
          % That is why it says Andys instead of Andy's.

% To get a hard copy of the plot type  the single word  >print   at the
% MATLAB prompt some time after you see the plot that you like



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% Put the block of code below (down to the line of %s)
% in   a file called   myfunction.m
% The first line of the file has to start with the word   function

function zdot = myfunction( t, z )   
% MYFUNCTION evaluates the right hand side of the ODE that is being
% solved.   Whatever name you use in the place of 'myfunction'
% must appear in 3 places:
%     1) in quotes as the first argument of ODE23
%     2) as the name of the file that contains the 
%        right hand side, namely   'myfunction.m', and
%     3) as the first word after the equal sign in the first
%        line of the  function that defines the right hand side
%        of the differential equation.

% ODE23 calls this function many times when finding its numerical
% solution to the differential equation.

global C   % this matches the global command in the script file 
           % above.  The value C can thus be used in this function
           % even though it is not expliticly passed.


zdot = C * sin ( z );   % this gives the rate of change of z
                        % in terms of its present value

end    % so long as you assign a value to the variable to the
       % left of the equal sign in the first line of the function
       % definition (in this case zdot) you don't need an end
       % statement.

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