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.

Dr. Kirti Khurdiya is an Assistant Professor in the Department of Mathematics and Statistics at Mohanlal Sukhadia University (MLSU), Udaipur, Rajasthan, India. She completed her M.Sc. in Mathematics from MLSU, where she was awarded the Gold Medal for securing the first position in the university. She also received the prestigious Dr. D. C. Gokhroo Medal from Rajasthan Ganita Parishad for obtaining the highest marks in M.Sc. Mathematics in Rajasthan. She qualified the CSIR–UGC NET in Mathematics and subsequently earned a Post Graduate Diploma in Business Administration (PGDBA) in Human Resource Management from Symbiosis, Pune. She obtained her Ph.D. in Mathematics in 2010, with research focused on inflationary and string cosmological models within the framework of General Relativity. Dr. Khurdiya has nearly two decades of teaching and research experience. She has published more than ten research papers in reputed national and international journals and has co-authored the book Advanced Matrices. Her research interests include cosmology, general relativity, and mathematical physics.

Mr. Bharat Kumar Yadav is an Assistant Professor in the Department of Mathematics and Statistics at Mohanlal Sukhadia University (MLSU), Udaipur, Rajasthan, India. He qualified the National Eligibility Test (NET) in Mathematics with an All-India Rank of 95, and the State Eligibility Test (SET) conducted by the Rajasthan Public Service Commission (RPSC), Ajmer. He is currently pursuing his Ph.D. in Mathematics at MLSU, Udaipur, with research interests in Graph Theory and Operations Research. He has published several research papers in reputed national and international journals. Mr. Yadav has more than eighteen years of teaching experience in Mathematics, Physics, and Computer Science. During his career, he has served in positions such as Lecturer, Senior Teacher, School Lecturer, and Assistant Professor, and has been associated with institutions including the College of Technology and Engineering (CTAE), Udaipur, the Department of Secondary Education, Bikaner, and the University of Rajasthan, Jaipur. He is recognized for his expertise in critical thinking, problem-solving, and collaborative teamwork.

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