This chapter focuses on solving ordinary differential equations (ODEs) using the Forward Euler method, both for single first-order ODEs and systems of first-order ODEs. It begins with a gentle introduction to ODEs through the example of population growth, where the differential equation $N'(t) = rN(t)$ is derived and solved using the Forward Euler method. The chapter then moves to more complex applications, such as oscillating mechanical systems, where the Forward Euler method is less effective and other methods like the Euler-Cromer scheme, Runge-Kutta schemes, and finite difference schemes are introduced. The chapter emphasizes the importance of initial conditions and provides a detailed derivation of the differential equation model for population growth, including the logistic model, which accounts for resource limitations. It also discusses the verification of numerical solutions through exact linear solutions and compares the accuracy of different time steps. In the section on the spreading of diseases, the chapter presents a compartment model (SIR model) for disease dynamics, where individuals are categorized into susceptible (S), infected (I), and recovered/immune (R) groups. The Forward Euler method is applied to this system of ODEs, and the chapter includes a Python implementation to simulate the spread of a flu in a closed society. The chapter concludes with an extension to the SIR model to account for time-restricted immunity, demonstrating how parameters affect disease dynamics.This chapter focuses on solving ordinary differential equations (ODEs) using the Forward Euler method, both for single first-order ODEs and systems of first-order ODEs. It begins with a gentle introduction to ODEs through the example of population growth, where the differential equation $N'(t) = rN(t)$ is derived and solved using the Forward Euler method. The chapter then moves to more complex applications, such as oscillating mechanical systems, where the Forward Euler method is less effective and other methods like the Euler-Cromer scheme, Runge-Kutta schemes, and finite difference schemes are introduced. The chapter emphasizes the importance of initial conditions and provides a detailed derivation of the differential equation model for population growth, including the logistic model, which accounts for resource limitations. It also discusses the verification of numerical solutions through exact linear solutions and compares the accuracy of different time steps. In the section on the spreading of diseases, the chapter presents a compartment model (SIR model) for disease dynamics, where individuals are categorized into susceptible (S), infected (I), and recovered/immune (R) groups. The Forward Euler method is applied to this system of ODEs, and the chapter includes a Python implementation to simulate the spread of a flu in a closed society. The chapter concludes with an extension to the SIR model to account for time-restricted immunity, demonstrating how parameters affect disease dynamics.