How can I solve this coupled non-linear differential equation and graph it?

I have two coupled non-linear differential equations. Please guide me how to solve it using MATLAB. I am a beginner.

a and n are constants. The initial condition are rho = a, omega = 0 and theta = 0. I also need to plot the graph.

Answer 1

[Sam Chak]

Dear @Avneet

You can start with something like the following. Let us know if the MATLAB code is helpful for a head start.

a       = 1;
tspan   = [0 10];       % time span
initv   = [0 a];        % initial values
[t, x]  = ode45(@odefcn, tspan, initv);
plot(t, x, 'linewidth', 1.5), grid on, xlabel('\theta'), legend('\omega', '\rho', 'location', 'best', 'FontSize', 14)

% Type the nonlinear coupled differential equations here
function dxdt = odefcn(t, x)
    dxdt    = zeros(2, 1);
    a       = 1.0;                      % constant
    n       = 0.5;                      % constant
    dxdt(1) = a*cos(x(1) - t)/x(2);     % Eqn 1
    dxdt(2) = a*(sin(x(1) - t) - n);    % Eqn 2
end

[Avneet]

Thanks!

Answer 2

[Chunru]

% initial condition cannot be [0, 0] since when rho0=0, you cannot define
% d omega / d theta.

[theta, y] = ode45(@myODE, [0 2*pi], [0 0.1]);

plot(theta, y)

function dy = myODE(theta, y)
    a = 1; n= 1;
    % y(1) -> omega,  y(2) -> rho
    dy(1, 1) = a*cos(y(1)-theta)/y(2);
    dy(2, 1) = a*(sin(y(1)-theta)-n);
end