Question

Problem 1. (15 points) Solve the following system of ODEs using your Euler implementation and ode45 and compare the errors at the final step. Use h 0.1 and 10 steps. What is the exact solution? Problem 2. (15 points) Express the following differential equation as a system of first order ODEs. Identify all critical points and identify their stability.

Answer 1

Problem 1:

Programming code in MATLAB

---------------------- ode45

---------------------- let
f=@(t,y) [ y(1)-y(2); -y(1)-y(2)];
tspn=[0 1];
y0=[0 4];
tol=10^(-6);
[t,ya]=ode45(f,tspn,y0,tol);
disp('using ode45 :');
disp(ya(end,:));

--------------------- Euler method
h=0.1;
t1=0:h:1;
%%% let x=y(1), y=y(2)
f1=@(x ,y) x-y;
f2=@(x ,y) -x-y;
x(1)=0;
y(1)=4;
for k=1:length(t1)
x(k+1)=x(k)+h*feval(f1,x(k),y(k));
y(k+1)=y(k)+h*feval(f2,x(k),y(k));
end

sol=[x(end) y(end)];
disp('using Eular :');
disp(sol);
err=sum((ya(end,:)-[x(end) y(end)]).^2);
fprintf('Error in vector : %f \n',err);

MATLAB O/P:

using ode45 :
-5.4732 3.2395

using Euler :
-5.7950 3.1479

Error in vector : 0.111941

---------------------- Exact solution

clc;
clear all;
close all;
syms s
A=[1 -1;-1 -1];
x0=[0;4];
tf=s*eye(2)-A;
gh=ilaplace(inv(tf)*x0);
subs(gh,1)
disp('Exact solution :');
disp(gh);
disp('at t=1');
sol=subs(gh,1);
fprintf('solution is[%f , %f ] : \n',sol(1),sol(2));

MATLAB O/P:

ans =

-2*2^(1/2)*sinh(2^(1/2))
4*cosh(2^(1/2)) - 2*2^(1/2)*sinh(2^(1/2))

Exact solution :
-2*2^(1/2)*sinh(2^(1/2)*t)
4*cosh(2^(1/2)*t) - 2*2^(1/2)*sinh(2^(1/2)*t)

at t=1
solution is[-5.473195 , 3.239539 ] :
>> err

Solut ion - iven that Sem f : U 0 3 30 Coo is ony

Pind eigen vaing of 3 2. 1 3 's unsob b