Foundation of Differential Equations and Their Applications presents a comprehensive introduction to ordinary and partial differential equations, emphasizing both theoretical foundations and practical applications. The book systematically explores first-order, higher-order, linear, nonlinear, and partial differential equations, together with classical analytical and computational methods used in modern mathematics, physics, and engineering. Important techniques such as integrating factors, complementary functions, variation of parameters, Fourier and Laplace transforms, Charpit’s and Monge’s methods, and numerical approaches are discussed in a clear and structured manner. The text further highlights real-world applications involving heat transfer, wave propagation, fluid flow, oscillations, turbulence, and nonlinear dynamic systems. Designed for undergraduate and postgraduate students, educators, and researchers, the book combines conceptual clarity with worked examples, summaries, and practice questions to strengthen understanding. It serves as a valuable resource for developing analytical skills and appreciating the role of differential equations in scientific and technological advancement.

Chapter 1. Fundamental Concepts of Differential Equations

Chapter 2. First-Order and First-Degree Differential Equations

Chapter 3. Higher Order Linear Differential Equations with Constant Coefficients

Chapter 4. Homogeneous Linear Differential Equations of Higher Order

Chapter 5. Second Order Linear Differential Equations

Chapter 6. Fundamentals of First-Order Linear Partial Differential Equations

Chapter 7. Fundamentals of First-Order Nonlinear Partial Differential Equations

Chapter 8. Linear Partial Differential Equations of the Second Order with Variable Coefficients

Chapter 9. Insights into the Dynamics of Linear and Nonlinear Partial Differential Equations