function a2sol  %insert the exact file name (as explained in the submission
                           % instructions) after "function"

%------------------------------------------------------------------------
%%%               ENGINEERING MATHEMATICS III - MATH 2Z03
%%%                             ASSIGNMENT #2
%%%             EULER's METHOD FOR INITIAL--VALUE PROBLEMS
%------------------------------------------------------------------------


%------------------------------------------------------------------------
% Covers:
% - "Numerical Mathematics" by M. Grasselli and D. Pelinovsky,
%    Sections 9.1.
% - "Advanced Engineering Mathematics" by D.G. Zill and W.S. Wright,
%    (Jones and Bartlett, 4th edition) Sections 2.6, 2.7, 2.8.
%------------------------------------------------------------------------


%------------------------------------------------------------------------
% 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/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('  ')
% Insert here your solution to Question #1
% (if you define your own functions, they may appear 
%  at the end of the file)

a=6;
b=7;
h = 0.05; % time step
y0 = 1+b/10; % initial value of y(0)
T = 1; % length of interval [a,a+1]
n = T/h; % number of steps
x(1) = 0; y(1) = y0; % values of t and y at first step k = 1
for k = 1:n % Loop to generate steps from k = 1 to k = n
  x(k+1) = x(k) + h; % increment for t (**)
  y(k+1) = y(k) + h*(2*x(k)*sin(x(k)^2))*(y(k)^2-(a+2)*y(k)); % update for the solution y (*)
end
Answer1=y(n+1);
h = 0.01; % time step
y0 = 1+b/10; % initial value of y(0)
T = 1; % length of interval [a,a+1]
n = T/h; % number of steps
x(1) = 0; z(1) =y0; % values of t and y at first step k = 1
for k = 1:n % Loop to generate steps from k = 1 to k = n
  x(k+1) = x(k) + h; % increment for t (**)
  z(k+1) = z(k) + h*(2*x(k)*sin(x(k)^2))*(z(k)^2-(a+2)*z(k)); % update for the solution y (*)
end
Answer2=z(n+1);




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)
r=a+2;
s=1+b/10;
y_ex=r.*s./((r-s).*exp(r.*(1-cos(x.^2)))+s);
y_num=z;
plot(x,y_ex,x,y_num,'*')
legend('y_{ex}', 'y_{num}','location','Northeast');
xlabel('x'); ylabel('y');
grid on; 
disp('  ')
disp('  ------------------------------------------------  ')  
disp('  ')
disp('  ')
disp('  ')
disp('  ')


disp('  ----------   Solution to Question #3  ----------  ')  
disp('  ')
% Insert here your solution to Question #2
% (if you define your own functions, they may appear 
%  at the end of the file)

c=8;
d=9;
k=2+c/10;
r=1.25+d/100;
Answer3=k^(1/(r-1));


h = 0.1;% time step
P0 = 1;% initial value of P(0)
t(1) = 0;
P(1)=P0;
j=1; 
q=1;
while abs(Answer3-P(j))>=0.5 

  t(j+1) = t(j) + h; % increment for t (**)
  P(j+1) = P(j)+h*(0.2)*(k*P(j)-P(j)^r);  % update for the solution P(*)
  if (P(j)<=Answer3/2) & (P(j+1)>Answer3/2)
      q=t(j);
  end
j=j+1;
end
Answer4=q
figure
plot(t,P)
xlabel('t'); ylabel('P');
grid on; 
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