function lab4soln  % 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 4';
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)

% reduce third order equation to first order system
% the "y" here is actually the column vector [y''; y'; y]
f = @(t,y) [2*t.^2+10*t+8; 0; 0] + [-4,5,-2; 1,0,0; 0,1,0]*y;

% this was missing from the original problem
a = 13;

% the 3rd column of Y1, Y2, etc, is the approximate solution
[T1,Y1] = ode23(f,[0 a],[3 -1 1]);
[T2,Y2] = ode45(f,[0 a],[3 -1 1]);
[T3,Y3] = ode113(f,[0 a],[3 -1 1]);
[T4,Y4] = ode15s(f,[0 a],[3 -1 1]);

figure('Name','Approximate solutions using 4 different solvers');
title({'Approximate solutions to:', ...
    'y''''''+ 4y''''+5y''+2y = 2t^2+10t+8', ...
    'y(0) = 1, y''(0) = -1, y''''(0) = 3'});
xlabel('t'); ylabel('y');
hold on;
plot(T1,Y1(:,3),T2,Y2(:,3),T3,Y3(:,3),T4,Y4(:,3));
legend('ode23','ode45','ode113','ode15s');
legend('Location','NorthWest');


% (b)

figure('Name','Timesteps used for 4 different solvers');
title({'Size of timesteps used by the indicated solvers in approximating solutions to', ...
    'y''''''+ 4y''''+5y''+2y = 2t^2+10t+8', ...
    'y(0) = 1, y''(0) = -1, y''''(0) = 3'});
xlabel('t'); ylabel('h (timestep)');
hold on;
plot(T1(2:end),T1(2:end)-T1(1:end-1), ...
    T2(2:end),T2(2:end)-T2(1:end-1), ...
    T3(2:end),T3(2:end)-T3(1:end-1), ...
    T4(2:end),T4(2:end)-T4(1:end-1));
legend('ode23','ode45','ode113','ode15s');
legend('Location','NorthWest');



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








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

Name 
Surname 
ID




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