give an example of solving an ordinary differential equation system with 3 equations in matlab

Here is an example of solving a system of ordinary differential equations (ODEs) with 3 equations using MATLAB:

main.m
% Define the system of ODEs
function dydt = odesystem(t, y)
    % Parameters
    a = 0.5;
    b = 0.25;

    % Variables
    y1 = y(1);
    y2 = y(2);
    y3 = y(3);

    % System of ODEs
    dydt = [
        -a*y1 + b*y2*y3;
        a*y1 - b*y2*y3;
        b*y2*y3
    ];
end

% Set initial conditions and time span
y0 = [1; 1; 1];
tspan = [0, 10]; % Start and end times

% Solve the ODE system
[t, y] = ode45(@odesystem, tspan, y0);

% Plot the results
plot(t, y(:, 1), 'r-', t, y(:, 2), 'g--', t, y(:, 3), 'b:');
xlabel('Time');
ylabel('Solution');
legend('y1', 'y2', 'y3');
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In this example, we define a system of ODEs using a function odesystem. The function takes two arguments, t representing time and y representing the vector of dependent variables. Inside the function, we specify the system of ODEs and calculate the derivatives, which are returned as dydt.

We then set the initial conditions y0 and the time span tspan. The ode45 function is used to solve the ODE system, which returns the time vector t and the solution matrix y.

Finally, we plot the solutions over time using the plot function, with each solution y1, y2, y3 assigned a different color and line style.

This example showcases how to numerically solve a system of ODEs in MATLAB and plot the results. The specific equations and parameters can be modified according to your needs.