Introduction
Differential equations play a central role in mathematics and the applied sciences. They describe relationships between quantities and their rates of change, and therefore appear naturally in the modeling of physical, biological, and economic systems. Examples include the motion of objects, population growth, heat diffusion, wave propagation, and electrical circuits. In this chapter, we begin by studying ordinary differential equations (ODEs), focusing on first- and second-order equations and the classical methods used to solve them. We then introduce the basic concepts of partial differential equations (PDEs), which involve functions of several variables and arise in more complex phenomena such as fluid flow and heat conduction. The goal is not only to compute solutions but also to understand how differential equations are formulated, interpreted, and applied to real-world situations.
Chapter Objectives
At the end of this chapter, students should be able to:
1. Understand the concept of a differential equation and its role in modeling physical and natural phenomena.
2. Solve first-order ordinary differential equations using methods such as separation of variables, integrating factors, and exact equations.
3. Solve second-order linear differential equations with constant coefficients.
4. Analyze the behavior of solutions, including existence, uniqueness, and qualitative interpretation.
5. Recognize basic forms of partial differential equations and understand the role of boundary and initial conditions.
6. Apply differential equations to practical problems in physics, engineering, and other sciences.
