function lab2soln  % where Name_XXXXXXX_hwN is the name of the file you submit without the extension .m

%------------------------------------------------------------------------
%%%               ENGINEERING MATHEMATICS IV - MATH 2ZZ3
%%%                             ASSIGNMENT #5:
%%%       TRIPLE INTEGRALS, DIVERGENCE THEOREM AND CHANGE OF VARIABLES
%------------------------------------------------------------------------


%------------------------------------------------------------------------
% Covers:
% - "Numerical Mathematics" by M. Grasselli and D. Pelinovsky,
%    Sections 7.7--7.8.
% "Advanced Engineering Mathematics" by D.G. Zill and W.S. Wright, section 9.15, 9.16, 9.17
%------------------------------------------------------------------------

%------------------------------------------------------------------------
% Instructions:
% - Submit your assignment electronically (via Email) to the
%   address specific to your last name 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/MATH2ZZ3/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 = 'Lab 2';
Surname = 'Solutions';
ID = 1234567;

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


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


disp('  ----------   Solution to Question #1  ----------  ')  
disp('  ')

% (a)

t1 = 0:0.1:3;
t2 = 0:0.01:3;

% initialize output arrays for better efficiency
y1 = zeros(size(t1));
y2 = zeros(size(t2));

for n = 2:31
    y1(n) = y1(n-1) + 0.1*(-y1(n-1)+t1(n-1));
end

Answer1 = y1(end);

for n = 2:301
    y2(n) = y2(n-1) + 0.01*(-y2(n-1)+t2(n-1));
end

Answer2 = y2(end);

% (b)

t3 = 0:0.01:3;
y3 = t3-1+exp(-t3);

figure('Name','Euler''s method and exact solution for y'' = -y + t');
title('Euler''s method and exact solution for y'' = -y + t');
hold on;
plot(t1,y1,t2,y2,t3,y3);
legend('Euler: h = 0.1','Euler: h = 0.01','exact solution');
legend('Location','NorthWest');


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


disp('  ----------   Solution to Question #2  ----------  ')  
disp('  ')

% (a) and (b)

t = 0:0.01:100; % everything will happen within 100 years ... probably
P = zeros(size(t));
P(1) = 50;

n = 1;
while P(n) < 300
    n = n + 1;
    P(n) = P(n-1) + 0.01*( exp(-t(n-1))*P(n-1) - P(n-1)*(P(n-1)-300)/200 );
    P(n)
end

Answer3 = t(n);
    
while P(n) >= P(n-1)
    n = n + 1;
    P(n) = P(n-1) + 0.01*( exp(-t(n-1))*P(n-1) - P(n-1)*(P(n-1)-300)/200 );    
end

Answer4 = t(n-1);
Answer5 = P(n-1);


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
Answer3
Answer4
Answer5




% User's Functions Space is below --------------------------