5. First-Order Differential Equations

This chapter delves into the world of first-order differential equations, which are pivotal in modeling phenomena such as population growth, radioactive decay, and heat flow. It begins with an introduction to the general form of these equations and their significance in various scientific and engineering fields. The chapter then explores different types of first-order equations, including separable, linear, and exact equations, each with specific methods for finding solutions. A key focus is on solving linear differential equations using integrating factors, with detailed steps and examples provided. The chapter also highlights the importance of certain integrals and demonstrates their application through solved examples. MATLAB is utilized to verify results, adding a practical dimension to the theoretical explanations. Additionally, the chapter covers more complex equations involving trigonometric and hyperbolic functions, showcasing the versatility of first-order differential equations. The inclusion of exercises and initial conditions further enhances the learning experience, making this chapter a comprehensive guide for professionals seeking to master first-order differential equations.