function Name_XXXXXXX_hwN %insert the exact file name (as explained in the submission
                           % instructions) after "function"


%------------------------------------------------------------------------
%%%               ENGINEERING MATHEMATICS III - MATH 2Z03
%%%                             ASSIGNMENT #4
%%%             HOMEWORK \#4: HIGHER ORDER ODEs: SPRING -- MASS SYSTEM
%------------------------------------------------------------------------


%------------------------------------------------------------------------
% Covers:
% - "Numerical Mathematics" by M. Grasselli and D. Pelinovsky,
%    Sections 9.2.
% - "Advanced Engineering Mathematics" by D.G. Zill and W.S. Wright,
%    (Jones and Bartlett, 4th edition) Section 3.8
%------------------------------------------------------------------------


%------------------------------------------------------------------------
% Instructions:
% - Submit your assignment electronically (via Email) to the
%   address specific to your lab section as indicated on the course
%   website; hardcopy submissions will not be accepted.
% - It is obligatory to use the current MATLAB template file available at
%   http://www.math.mcmaster.ca/gabardo/M2Z03/frames/template.m; 
%   submissions non compliant with this template will not be accepted.
% - Make sure to enter your name and student I.D. number in the
%   appropriate section of the template.
% - Late submissions and submissions which do not comply with
%   these guidelines will not be accepted.
% - All graphs should contain suitable titles and legends.
%------------------------------------------------------------------------
% Written by Vladislav Bukshtynov, 2009

clc; close all; clear all;

% Student information
  
disp('  ------------   Student Information  ------------  ')  

% Please enter your information here
Name = 'Johny';
Surname = 'Good';
ID = 123456789;

fprintf('  Student:  %s %s (ID: %d) \n', Name, Surname, ID);



disp('  ------------------------------------------------  ')  
disp('  ')
disp('  ')
disp('  ')
disp('  ')


disp('  ----------   Solution to Question #1  ----------  ')  
disp('  ')
a=6;b=7;c=8;d=9;
% Insert here your solution to Question #1
% (if you define your own functions, they may appear 
%  at the end of the file)
h = 0.05; % time step for x value
x0 = 1; % initial value of y(0)
u0 = 1; % initial value of u(0)
T = 10;% maximum value of x
n = T/h; % number of steps
t(1) = 0; x(1) = x0; u(1) = u0;
for k = 1:n
  t(k+1) = t(k) + h; % increment for x
  x(k+1) = x(k) + h * u(k); % update for the solution y (5*)
  u(k+1) = u(k) + h *(-(0.5+0.1*a)*u(k)-(4+0.1*b)*x(k)); % update for the solution u (5**)
end
figure;
plot(t,x);
legend('x=x(t)');
xlabel('t'); ylabel('x');
grid on; 
i=0;
for k = 1:n
if u(k)*u(k+1)<= 0
i=i+1;
z(i)=t(k);
r(i)=x(k);
end
end
Answer1=[z;r]
disp('  ')
disp('  ------------------------------------------------  ')  
disp('  ')
disp('  ')
disp('  ')
disp('  ')

disp('  ----------   Solution to Question #2  ----------  ')  
disp('  ')
% Insert here your solution to Question #2
% (if you define your own functions, they may appear 
%  at the end of the file)

h = 0.001; % time step for x value
x0 = (1+c)/10; % initial value of y(0)
u0 = -d/10; % initial value of u(0)
T = 20;% maximum value of x
n = T/h; % number of steps
t(1) = 0; 
for j=1:6
    x(j,1) = x0; u(j,1) = u0;
    m(1,j)=x0;m(2,j)=x0
end
for k = 1:n
  t(k+1) = t(k) + h; % increment for x
  for j=1:6
  x(j,k+1) = x(j,k) + h * u(j,k); % update for the solution y (5*)
  u(j,k+1) = u(j,k) + h *(-16*x(j,k)+cos((3+(j-1)*0.2)*t(k))); % update for the solution u (5**)
       if x(j,k+1) > m(1,j)
      m(1,j)=x(j,k+1);
       end
      if x(j,k+1)<m(2,j)
      m(2,j)=x(j,k+1);
       end
  end
end
figure;grid on;
for l=1:6

subplot(2,3,l);
plot(t,x(l,:));
%legend('x=x(t) for k='l-1);
legend(sprintf('x=x(t) for k=%d',l-1)) 
xlabel('t'); ylabel('x');    
grid on;
end



Answer2=m


disp('  ')
disp('  ------------------------------------------------  ')  
disp('  ')
disp('  ')
disp('  ')
disp('  ')

disp('  ----------            Answers         ----------  ')  
disp('  ')
% DO NOT TOUCH THIS PART (RESERVED FOR THE INSTRUCTOR'S USE)

Name 
Surname 
ID

Answer1
Answer2