Question

Draw a direction field using MATLAB to sketch a few of the trajectories, and describe the behavior of the solutions as t → ∞

x' = [ 1 -1 ]

[ 5 -3 ] x

It's a matrix, sorry its bad, so x' = [ ] *x

I just do not know how to put this in Matlab, thanks

Answer 1

Matlab code for the problem

clc; clear; % Clearing the command window and workspace

[x,y] = meshgrid(-5:0.2:5); % Creating mesh grid of the domain

% Rewritting the matrix equation as system of equations ie, x = [x, y]' and

% dx/dt = [u, v]';

u = x-y; % dx/dt = x-y

v = 5*x-3*y; % dy/dt = 5x-3y

quiver(x,y,u,v); % plotting the vector field

hold on; % holding the figure to plot solutions

% To understand the nature of the solution plotting some solution to the

% ODE

t = [0 50]; %Time period

f = @(t,y) [y(1)-y(2); 5*y(1)-3*y(2)]; % The RHS of ODEs

initial = [3 1]; % Initial condition

[t,Y] = ode45(f,t,initial); %Computing the solution using ODE45 function

plot(Y(:,1),Y(:,2)); % plotting the solution for the initial condition [x, y] = [3 1]

initial = [-3 1]; % Initial condition

[t,Y] = ode45(f,t,initial); %Computing the solution using ODE45 function

plot(Y(:,1),Y(:,2)); % plotting the solution for the initial condition [x, y] = [-3 1]

initial = [-3 -1]; % Initial condition

[t,Y] = ode45(f,t,initial); %Computing the solution using ODE45 function

plot(Y(:,1),Y(:,2)); % plotting the solution for the initial condition [x, y] = [-3 -1]

initial = [3 -1]; % Initial condition

[t,Y] = ode45(f,t,initial); %Computing the solution using ODE45 function

plot(Y(:,1),Y(:,2)); % plotting the solution for the initial condition [x, y] = [3 -1]

xlabel('x'),ylabel('y'); % labeling the axis

Screen Print of the code

1 2 3 4 clc; clear; % Clearing the command window and workspace [x,y] = meshgrid(-5:0.2:5); % Creating mesh grid of the domain % Rewritting the matrix equation as system of equations ie, x = [x, y]' and % dx/dt = [u, v]'; u = x-Y; % dx/dt = x-y v = 5*X-3*y; % dy/dt = 5x-3y quiver(x, y, u, v); % plotting the vector field hold on; % holding the figure to plot solutions 5 - 7 8 9 10 11 12 - 13 14 15 16 - 17 18 - 19 20- 21 22 - % to understand the nature of the solution plotting some solution to the % ODE t = [050]; %Time period f = e(t,y) [y(1)-y(2); 5*y(1)-3*y(2)]; % The RHS of ODES initial = [3 1]; % Initial condition [t, Y] = ode45(f, t, initial); Computing the solution using ODE45 function plot(Y(:,1), Y(:,2)); % plotting the solution for the initial condition [x, y] = [3 1] initial = [-3 1]; % Initial condition [1,Y] = ode45(f, t, initial); %computing the solution using ODE45 function plot(Y(:,1),y(:,2)); % plotting the solution for the initial condition [x, y] = [-3 1] initial = [-3 -1]; % Initial condition|| [T,Y] = ode45(f, t, initial); %computing the solution using ODE45 function plot(Y(:,1), Y(:,2)); % plotting the solution for the initial condition [x, y] = [-3 -1] initial = [3 -1]; % Initial condition [t, Y] = ode45(f, t, initial); %Computing the solution using ODE45 function plot(Y(:,1),y(:,2)); % plotting the solution for the initial condition [x, y] = [3 -1] xlabel('x'),ylabel('y'); % labeling the axis 23- 24 25 26 27 - 28 - 29 30

Screen Print of the output

Figure 1 + 9 2 у 2. 4 -6 -6 -2 2 6 х

Solution tends to zero as t tends to infinity.