machinename:~$ matlab
>>
>> diary first-try	%creates a file named 'first-try' 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	We solve the initial value problem for first order ode
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%	x' + x = 5sin(2t)
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%	x(0)=1
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%	First, we plot the directional fields for the equation,
%       then the solution curve corresponding to the initial data,
%	and finally we plot our computed solution:
%		y(t) = sin(2t)-2cos(2t) + 3exp(-t)
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%       Both curves must be exactly the same!   
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

>>  dfield             %    x'= -x + 5*sin(2*t)

%       you'll get a figure window, with an equation x'= x^2-t. 	
%       Enter the equation  x'= - x + 5*sin(2*t) instead. 
%	Click on "proceed"
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%	Click on "file --> print" to get in
%       the directional field into you local printer.  
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%	To get the corresponding vector field  click on 
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%	Click on "print" to get  a plot of the vector field.
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%       Next   click on 
%	"Options --Keyboard input" and enter the initial data
%       t=0, x=1. Click on "compute" to get the solution curve 
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%	Click on "print" to get the solution (UNIQUE!) to the 
% 	initial value 
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%	Now you are going to plot your solution formula, 
%       previously computed by yourself, and check that 
%       coincides with the one of the Matlab dfield software:
%    go back to the window where the MATLAB prompt >> is, and type

>> t=linspace(-2,10);  (ENTER)

%	see that -2 and 10 are the bounds for t in the  
%	dfield5 display window.
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%	next type your solution with the MATLAB grammar:
 
 >>y= sin(2*t)-2*cos(2*t) + 3*exp(-t);    (ENTER)

%	next type

>> plot(t,y,'*')  (ENTER)

%	 A new curve made of stars '*' is plotted on dfield 
%	 display window. 
%	 Compare both graphs.
%	 If your solution is correct you should see (in principle)
%	 NO DIFFERENCE with the one you obtained by 
%	 entering the initial data on  dfield. 
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%	 to label your graph use "title" command as follows

>>  title('dx/dt + x = 5sin(2t), x(0)=1, by M.E.')

%	Click on "print" to get your graph. 
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%  keep in mind that these are numerical computations, so eventually 
%   THERE ARE DISCREPANCIES depending on the scale of your graph.
%	We shall address these issues later on.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%       You can also plot any function you wish, without using
%       dfield5.     
%	For instance, just  plot your computed solution function.
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%	First quit  dfield by clicking "quit" on the   display window.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

>> t=linespace(-2,10);  
>> y= sin(2*t)-2*cos(2*t) + 3*exp(-t);  

%	do not need to type the above sentences again, if you 
%	already have used dfield to plot this function,
%       however repeating them, it does not matter.

  
>> plot(t,y,'*')        % or use plot(t,y) for a solid line curve.
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%	you'll see the graph of y(t) in a different scale 
%	(chosen by matlab for this particular plot of this function)
%       In order to get the same scale as in the plot of dfield
%	you need to type
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>> axis([-2 10 -4 4])    %Change the y-axis to the values of the 
                         % dfield display.

%	label your graph
                      
>> title('Figure 1: y(t) = sin(2t)-2cos(2t) + 3exp(-t) by M.E.')    
%
% 	label the axis

>> xlabel('Independent Variable t')     %label horizontal axis
>> ylabel('Dependent variable y')        %label vertical axis
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% 	and you may want to add a grid  
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>> grid

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%	Print your plot by  "file-->print"
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%                DO NOT  FORGET!!!
>> diary off
>> quit           %(to properly quit 'matlab')

---------------------------------------------------------------------



	Your matlab executions of this problem are simply:
	(see that I did not typed twice  t=linspace(-2,10);
	 	and y= sin(2*t)-2*cos(2*t) + 3*exp(-t))
 

>> dfield     %    x'= -x + 5*sin(2*t)
>> t=linspace(-2,10);
>> y= sin(2*t)-2*cos(2*t) + 3*exp(-t);
>> plot(t,y,'*')
>> title('dx/dt + x = 5sin(2t), x(0)=1, by M.E.')
>> plot(t,y,'*')
>> axis([-2 10 -4 4])
>> title('Figure 1: y(t) = sin(2t)-2cos(2t) + 3exp(-t) by M.E.')
>> xlabel('Independent Variable t')
>> ylabel('Dependent variable y')
>> grid
>> diary off

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