📙 Chapter 3: Solving Ordinary Differential Equations (ODEs)

Ordinary Differential Equations (ODEs) are equations that involve an unknown function and its derivatives. They are fundamental in modeling a wide range of phenomena in physics, engineering, biology, and economics. While some ODEs can be solved analytically, many real-world problems lead to equations that cannot be solved exactly, making numerical methods essential.

Historically, the study of differential equations began in the 17th century with Isaac Newton and Gottfried Wilhelm Leibniz, who developed the foundations of calculus. In the 18th and 19th centuries, mathematicians such as Leonhard Euler, Joseph-Louis Lagrange, and Carl Gustav Jacobi developed methods to approximate solutions for practical problems, giving rise to the numerical methods we use today.

In this chapter, we focus on numerical techniques for solving ODEs. These methods allow us to approximate the solution of an ODE at discrete points when an exact solution is not available. We will study both single-step and multi-step methods, understand their convergence and stability properties, and analyze the trade-offs between accuracy and computational cost.

Numerical methods covered include:

• The bisection method for finding roots, fundamental in solving implicit ODE schemes.
• The Newton–Raphson method for nonlinear systems arising in ODE discretization.
• Other iterative schemes such as Euler’s method, Runge-Kutta methods, and predictor-corrector methods.

By the end of this chapter, students will understand the theory behind numerical ODE solvers, learn to implement them, analyze their accuracy, and apply them to real-world problems in science and engineering.