Question

Would someone help me with the last 3 questions here, I got the previous ones but need help with the last 3 in red. Thank you

3. a) Find the solution y(t) of the ordinary differential equation with the initial conditions Solve it only by hand and show your complete work. Do not use a calculator or any symbolic calu lations). [8 marks) b) i) Recast our third order ODE into a system of first order ODEs of the form v - A.v, where v, dv/d.r f(v) and v = (y, y,"y")T You should show all working to find the corresponding matrix A. Do not solve the system. 4 marks ii) Show your full work to find the initial condition v(0) to complete the initial value problem for the system of first order ODEs. 2 marks ii) Now that you find a first order system of ODEs, you can solve it using the Huens method ac- cording to the material you covered in computational module 2. A template of the code which will help you to get started is also available. Attach the complete code as a pdf page to your assignment. 4 marks ii) Plot the result of your code y(x)n Matlab, label the axes and the curve, attach the plot to your assignment. 1 mark] iv) Explain the influence of the initial conditions on the behaviour of y(t). Hint: You can change some of initial conditions in the code to find out how this is relevant for the behaviour of the solution.1 mark

Transcribed:

Now that you nd a rst order system of ODEs, you can solve it using the Huens method ac-
cording to the material you covered in computational module 2. A template of the code which
will help you to get started is also available. Attach the complete code as a pdf page to your
assignment. [4 marks]

Plot the result of your code y(x) in Matlab, label the axes and the curve, attach the plot to your
assignment. [1 mark]

Explain the in uence of the initial conditions on the behaviour of y(t). Hint: You can change
some of initial conditions in the code to nd out how this is relevant for the behaviour of the
solution. [1 mark]

Answer 1

4 (r-t 3 t- 乇 Эне 2

Le 2 O O 1 And iitra valiue.v(o) 3ツ

SMatlab code for Heun's method clear all close all dt-0.01i tmax-20; t-[0:dt:tmax ] N-length(t) v zeros (N, 3) y( 1.. )一0 5-39] ; A [0 1 0:0 0 1-124-31; for i-2:N K1-dt*A* (v(i-1,:))'; end yay ( : , 1) ; plot (t,y, 'Linewidth',2) Exact solution yy-l(tt) -3*exp(-3*tt)+3*cos (2*tt)-2*sin (2*tt) hold on plot(t,yy(t), '-') legend(Heun Method", 'Exact solution') title( 'Solution of y(x) using Heun method) xlabel ( ·x' ) ylabel (y(x) grid on 868888888888888888 End of Code 웅8888888888888888

Solution of y(x) using Heun method Heun Method 一一Exact solution -3 0 24 6 8 10 1 14 16 820 Published with MATLAB R2018a

%%Matlab code for Heun's method
clear all
close all

dt=0.01;
tmax=20;
t=[0:dt:tmax];
N=length(t);

v=zeros(N,3);
v(1,:)=[0 5 -39];

A=[0 1 0;0 0 1;-12 -4 -3];

for i=2:N
  
    K1=dt*A*(v(i-1,:))';
    K2=dt*A*((v(i-1,:)+K1'))';
  
    v(i,:)=v(i-1,:)+(K1'+K2')/2;
  
end

y=v(:,1);

plot(t,y,'Linewidth',2)

%Exact solution
yy=@(tt) -3*exp(-3*tt)+3*cos(2*tt)-2*sin(2*tt);

hold on
plot(t,yy(t),'--')
legend('Heun Method','Exact solution')
title('Solution of y(x) using Heun method')
xlabel('x')
ylabel('y(x)')

grid on

%%%%%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%%%%