**Course Title:** MATLAB Programming: Applications in Engineering, Data Science, and Simulation **Section Title:** Solving Differential Equations with MATLAB **Topic:** Solve a set of ODEs and visualize the results using MATLAB's built-in solvers. **Lab Objective:** In this lab, you will learn how to solve a set of Ordinary Differential Equations (ODEs) using MATLAB's built-in solvers and visualize the results. By the end of this lab, you will be able to: * Define a system of ODEs in MATLAB * Use MATLAB's built-in solvers to solve the system of ODEs * Visualize the results using MATLAB's plotting tools **Theory and Background:** A system of ODEs is a set of equations that describes the behavior of a dynamic system over time. In this lab, we will focus on systems of first-order ODEs, which can be written in the following form: dx/dt = f(t,x) where x is a vector of state variables, t is the independent variable (time), and f(t,x) is a vector of functions that describe the behavior of the system. **MATLAB's Built-in Solvers:** MATLAB provides several built-in solvers for solving systems of ODEs. The most common solvers are: * `ode45`: A non-stiff solver that uses a Runge-Kutta algorithm * `ode23`: A non-stiff solver that uses a Runge-Kutta algorithm * `ode113`: A stiff solver that uses a variable-order algorithm * `ode15s`: A stiff solver that uses a variable-order algorithm **Step-by-Step Instructions:** 1. **Define the System of ODEs:** Create a new MATLAB script and define the system of ODEs using a function handle. For example, let's consider the following system of ODEs: dx/dt = [-0.5*x(1) + x(2)*sin(t)] [0.5*x(1)*cos(t) - 0.5*x(2)] This system can be defined in MATLAB as follows: ```matlab function dxdt = myode(t,x) dxdt = [-0.5*x(1) + x(2)*sin(t); 0.5*x(1)*cos(t) - 0.5*x(2)]; end ``` 2. **Solve the System of ODEs:** Use MATLAB's built-in solver `ode45` to solve the system of ODEs. You will need to specify the following inputs: * `tspan`: A vector of time points where the solution is desired * `x0`: The initial condition of the system * `odefun`: The function handle that defines the system of ODEs For example: ```matlab [t,x] = ode45(@myode, [0 10], [1; 0]); ``` This will solve the system of ODEs over the time interval [0 10] with the initial condition x(0) = [1; 0]. 3. **Visualize the Results:** Use MATLAB's plotting tools to visualize the results. For example: ```matlab plot(t,x(:,1)); xlabel('Time (s)'); ylabel('x_1'); ``` This will plot the solution of the first state variable x_1(t) over time. **Exercise:** 1. Define a system of ODEs using a function handle. 2. Use MATLAB's built-in solver `ode45` to solve the system of ODEs. 3. Visualize the results using MATLAB's plotting tools. **Additional Resources:** * MATLAB Documentation: [Ordinary Differential Equations](https://www.mathworks.com/help/matlab/ordinary-differential-equations.html) * MATLAB Documentation: [ode45](https://www.mathworks.com/help/matlab/ref/ode45.html) **Do you have any questions about this lab? Leave a comment below.** **What's Next:** In the next topic, we will introduce optimization in MATLAB using `fminsearch`, `fmincon`, etc. This will be a critical tool for solving optimization problems in engineering, data science, and simulation.