Numerical solution of differential equations is one of the most widely applied areas in numerical analysis. In fields such as fluid mechanics, heat transfer, and structural analysis, we frequently encounter ordinary differential equations (ODEs), systems of ODEs, or more generally, partial differential equations (PDEs).

For example, the pendulum problem discussed in Chapter 1 will be solved here numerically. The numerical solution can then be analyzed and compared with other approximate or quasi-analytical solutions. Numerical methods allow us to study complex problems for which exact analytical solutions are not available but are practically important.

In this chapter, we begin with relatively simple methods that have a clear geometric interpretation and gradually progress to more advanced techniques, such as fourth-order Runge-Kutta methods, which provide highly accurate results. We mainly consider differential equations with initial conditions, but we will also briefly explore boundary value problems using shooting and finite difference methods.

The starting point is the general form of a first-order differential equation with an initial condition. The task is to determine a function y(t) that satisfies:

{
  y′(t) = f(t, y(t))
  y(t₀) = y₀
}

Here, t is the independent variable, often representing time, and y is the dependent variable. The function f is assumed to be sufficiently differentiable. The initial condition y(t₀) = y₀ represents the state of the system at the starting point. The goal is to find y(t) for t ≥ t₀, either analytically or using a numerical approximation.

Numerical methods allow us to approximate solutions to ODEs when analytical formulas are unavailable, enabling practical analysis and simulations in engineering and scientific applications.