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

%------------------------------------------------------------------------
%%%               ENGINEERING MATHEMATICS III - MATH 2Z03
%%%                             ASSIGNMENT #5
%%%             HOMOGENEOUS SYSTEMS OF LINEAR ODEs
%------------------------------------------------------------------------


%------------------------------------------------------------------------
% Covers:
% - "Numerical Mathematics" by M. Grasselli and D. Pelinovsky,
%    Sections 1.6, 5.6,  9.2.
% - "Advanced Engineering Mathematics" by D.G. Zill and W.S. Wright,
%    (Jones and Bartlett, 4th edition) Section 10.2.
%------------------------------------------------------------------------


%------------------------------------------------------------------------
% 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('  ')
% 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;c=8;d=9;
A=[-(1+a/10), -(10+b);10+c, -(1+1/10)]
Answer1=eig(A)'
[r,p]=eig(A)
in=[real(p(1,1));imag(p(1,1))];
B=[in, real(r(:,1)), imag(r(:,1))] ;
Answer2=B
x0(:,1)=[1;-1]; x0(:,2)=[-1;-1];
D=[real(r(:,1)), imag(r(:,1))];
C=D^(-1)*x0
Answer3=C
t=[0:0.01:5];
for i=1:501
x(:,i)=exp(B(1,1)*t(i)).*(    (  cos(B(2,1)*t(i)).*B(:,2)-sin(B(2,1)*t(i)).*B(:,3)   ).*C(1,1)     +   (sin(B(2,1)*t(i)).*B(:,2)+cos(B(2,1)*t(i)).*B(:,3)).*C(2,1)    );
u(:,i)=exp(B(1,1)*t(i)).*(    (  cos(B(2,1)*t(i)).*B(:,2)-sin(B(2,1)*t(i)).*B(:,3)   ).*C(1,2)     +   (sin(B(2,1)*t(i)).*B(:,2)+cos(B(2,1)*t(i)).*B(:,3)).*C(2,2)    );
end
figure
grid on; 
plot(x(1,:), x(2,:), u(1,:), u(2,:))
grid on; 
xlabel('x'); ylabel('y');

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)

A=[1 2+a/10; 4+b/10 -1];
 [r,p]=eig(A)
h=0.01;
Answer4=[p(1,1),p(2,2)];

xmax=10;
ymax=10;


j=1;
Xforward(:,1)=[1;0];

while (abs(Xforward(1,j))<= xmax) & (abs(Xforward(2,j))<=ymax)
Xforward(:,j+1)=Xforward(:,j)+h.*A*Xforward(:,j);
j=j+1;
end

Xbackward(:,1)=[1;0];
k=1;

while (abs(Xbackward(1,k))<= xmax) & (abs(Xbackward(2,k))<=ymax)
Xbackward(:,k+1)=Xbackward(:,k)-h.*A*Xbackward(:,k);
k=k+1;
end
j=1;
Yforward(:,1)=[-1;0];

while (abs(Yforward(1,j))<= xmax) & (abs(Yforward(2,j))<=ymax)
Yforward(:,j+1)=Yforward(:,j)+h.*A*Yforward(:,j);
j=j+1;
end

Ybackward(:,1)=[-1;0];
k=1;

while (abs(Ybackward(1,k))<= xmax) & (abs(Ybackward(2,k))<=ymax)
Ybackward(:,k+1)=Ybackward(:,k)-h.*A*Ybackward(:,k);
k=k+1;
end
j=1;
Uforward(:,1)=[0;1];

while (abs(Uforward(1,j))<= xmax) & (abs(Uforward(2,j))<=ymax)
Uforward(:,j+1)=Uforward(:,j)+h.*A*Uforward(:,j);
j=j+1;
end

Ubackward(:,1)=[0;1];
k=1;

while (abs(Ubackward(1,k))<= xmax) & (abs(Ubackward(2,k))<=ymax)
Ubackward(:,k+1)=Ubackward(:,k)-h.*A*Ubackward(:,k);
k=k+1;
end

j=1;
Tforward(:,1)=[0;-1];

while (abs(Tforward(1,j))<= xmax) & (abs(Tforward(2,j))<=ymax)
Tforward(:,j+1)=Tforward(:,j)+h.*A*Tforward(:,j);
j=j+1;
end

Tbackward(:,1)=[0;-1];
k=1;

while (abs(Tbackward(1,k))<= xmax) & (abs(Tbackward(2,k))<=ymax)
Tbackward(:,k+1)=Tbackward(:,k)-h.*A*Tbackward(:,k);
k=k+1;
end

figure;
hold on;
grid on;
plot(Xforward(1,:),Xforward(2,:),'blue')
plot(Xbackward(1,:),Xbackward(2,:),'blue')
plot(Yforward(1,:),Yforward(2,:),'green')
plot(Ybackward(1,:),Ybackward(2,:),'green')
plot(Uforward(1,:),Uforward(2,:),'red')
plot(Ubackward(1,:),Ubackward(2,:),'red')
plot(Tforward(1,:),Tforward(2,:),'yellow')
plot(Tbackward(1,:),Tbackward(2,:),'yellow')
Eig1for(:,1)=[0;0];
i=1;
while (abs(Eig1for(1,i))<= xmax) & (abs(Eig1for(2,i))<=ymax)
Eig1for(:,i+1)=i.*r(:,1)*h;
i=i+1;
end
Eig1back(:,1)=[0;0];
j=1;

while (abs(Eig1back(1,j))<= xmax) & (abs(Eig1back(2,j))<=ymax)
Eig1back(:,j+1)=-j.*r(:,1)*h;
j=j+1;
end

plot(Eig1for(1,:),Eig1for(2,:), 'black')
plot(Eig1back(1,:),Eig1back(2,:), 'black')
Eig2for(:,1)=[0;0];
i=1;

while (abs(Eig2for(1,i))<= xmax) & (abs(Eig2for(2,i))<=ymax)
Eig2for(:,i+1)=i.*r(:,2)*h;
i=i+1;
end
Eig2back(:,1)=[0;0];
j=1;

while  (abs(Eig2back(2,j))<= xmax) & (abs(Eig2back(2,j))<=ymax)
Eig2back(:,j+1)=-j.*r(:,2)*h;
j=j+1;
end
plot(Eig2for(1,:),Eig2for(2,:), 'black')
plot(Eig2back(1,:),Eig2back(2,:), 'black')

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





% You can add your own user-defined functions below